imbi12VD

Description: Implication form of imbi12i . The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. imbi12 is imbi12VD without virtual deductions and was automatically derived from imbi12VD .

		Ref	Expression
	Assertion	imbi12VD	$⊢ (φ \leftrightarrow ψ) \to ((χ \leftrightarrow θ) \to ((φ \to χ) \leftrightarrow (ψ \to θ)))$

Proof

Step	Hyp	Ref	Expression
1		idn2	$⊢ ((φ \leftrightarrow ψ), (χ \leftrightarrow θ) \to (χ \leftrightarrow θ))$
2		idn1	$⊢ ((φ \leftrightarrow ψ) \to (φ \leftrightarrow ψ))$
3		idn3	$⊢ ((φ \leftrightarrow ψ), (χ \leftrightarrow θ), (φ \to χ) \to (φ \to χ))$
4		biimpr	$⊢ (φ \leftrightarrow ψ) \to (ψ \to φ)$
5	4	imim1d	$⊢ (φ \leftrightarrow ψ) \to ((φ \to χ) \to (ψ \to χ))$
6	2 3 5	e13	$⊢ ((φ \leftrightarrow ψ), (χ \leftrightarrow θ), (φ \to χ) \to (ψ \to χ))$
7		biimp	$⊢ (χ \leftrightarrow θ) \to (χ \to θ)$
8	7	imim2d	$⊢ (χ \leftrightarrow θ) \to ((ψ \to χ) \to (ψ \to θ))$
9	1 6 8	e23	$⊢ ((φ \leftrightarrow ψ), (χ \leftrightarrow θ), (φ \to χ) \to (ψ \to θ))$
10	9	in3	$⊢ ((φ \leftrightarrow ψ), (χ \leftrightarrow θ) \to ((φ \to χ) \to (ψ \to θ)))$
11		idn3	$⊢ ((φ \leftrightarrow ψ), (χ \leftrightarrow θ), (ψ \to θ) \to (ψ \to θ))$
12		biimp	$⊢ (φ \leftrightarrow ψ) \to (φ \to ψ)$
13	12	imim1d	$⊢ (φ \leftrightarrow ψ) \to ((ψ \to θ) \to (φ \to θ))$
14	2 11 13	e13	$⊢ ((φ \leftrightarrow ψ), (χ \leftrightarrow θ), (ψ \to θ) \to (φ \to θ))$
15		biimpr	$⊢ (χ \leftrightarrow θ) \to (θ \to χ)$
16	15	imim2d	$⊢ (χ \leftrightarrow θ) \to ((φ \to θ) \to (φ \to χ))$
17	1 14 16	e23	$⊢ ((φ \leftrightarrow ψ), (χ \leftrightarrow θ), (ψ \to θ) \to (φ \to χ))$
18	17	in3	$⊢ ((φ \leftrightarrow ψ), (χ \leftrightarrow θ) \to ((ψ \to θ) \to (φ \to χ)))$
19		impbi	$⊢ ((φ \to χ) \to (ψ \to θ)) \to (((ψ \to θ) \to (φ \to χ)) \to ((φ \to χ) \leftrightarrow (ψ \to θ)))$
20	10 18 19	e22	$⊢ ((φ \leftrightarrow ψ), (χ \leftrightarrow θ) \to ((φ \to χ) \leftrightarrow (ψ \to θ)))$
21	20	in2	$⊢ ((φ \leftrightarrow ψ) \to ((χ \leftrightarrow θ) \to ((φ \to χ) \leftrightarrow (ψ \to θ))))$
22	21	in1	$⊢ (φ \leftrightarrow ψ) \to ((χ \leftrightarrow θ) \to ((φ \to χ) \leftrightarrow (ψ \to θ)))$

1::	\|- (. ( ph <-> ps ) ->. ( ph <-> ps ) ).
2::	\|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ch <-> th ) ).
3::	\|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ph -> ch ) ).
4:1,3:	\|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ps -> ch ) ).
5:2,4:	\|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ph -> ch ) ->. ( ps -> th ) ).
6:5:	\|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ph -> ch ) -> ( ps -> th ) ) ).
7::	\|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ps -> th ) ).
8:1,7:	\|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ph -> th ) ).
9:2,8:	\|- (. ( ph <-> ps ) ,. ( ch <-> th ) ,. ( ps -> th ) ->. ( ph -> ch ) ).
10:9:	\|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ps -> th ) -> ( ph -> ch ) ) ).
11:6,10:	\|- (. ( ph <-> ps ) ,. ( ch <-> th ) ->. ( ( ph -> ch ) <-> ( ps -> th ) ) ).
12:11:	\|- (. ( ph <-> ps ) ->. ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) ).
qed:12:	\|- ( ( ph <-> ps ) -> ( ( ch <-> th ) -> ( ( ph -> ch ) <-> ( ps -> th ) ) ) )

Metamath Proof Explorer

Theorem imbi12VD

Proof