Metamath Proof Explorer

Theorem mdandyvrx3

Description: Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016)

		Ref	Expression
	Hypotheses	mdandyvrx3.1	$⊢ (φ ⊻ ζ)$
		mdandyvrx3.2	$⊢ (ψ ⊻ σ)$
		mdandyvrx3.3	$⊢ χ \leftrightarrow ψ$
		mdandyvrx3.4	$⊢ θ \leftrightarrow ψ$
		mdandyvrx3.5	$⊢ τ \leftrightarrow φ$
		mdandyvrx3.6	$⊢ η \leftrightarrow φ$
	Assertion	mdandyvrx3	$⊢ ((((χ ⊻ σ) \land (θ ⊻ σ)) \land (τ ⊻ ζ)) \land (η ⊻ ζ))$

Proof

Step	Hyp	Ref	Expression
1		mdandyvrx3.1	$⊢ (φ ⊻ ζ)$
2		mdandyvrx3.2	$⊢ (ψ ⊻ σ)$
3		mdandyvrx3.3	$⊢ χ \leftrightarrow ψ$
4		mdandyvrx3.4	$⊢ θ \leftrightarrow ψ$
5		mdandyvrx3.5	$⊢ τ \leftrightarrow φ$
6		mdandyvrx3.6	$⊢ η \leftrightarrow φ$
7	2 3	axorbciffatcxorb	$⊢ (χ ⊻ σ)$
8	2 4	axorbciffatcxorb	$⊢ (θ ⊻ σ)$
9	7 8	pm3.2i	$⊢ ((χ ⊻ σ) \land (θ ⊻ σ))$
10	1 5	axorbciffatcxorb	$⊢ (τ ⊻ ζ)$
11	9 10	pm3.2i	$⊢ (((χ ⊻ σ) \land (θ ⊻ σ)) \land (τ ⊻ ζ))$
12	1 6	axorbciffatcxorb	$⊢ (η ⊻ ζ)$
13	11 12	pm3.2i	$⊢ ((((χ ⊻ σ) \land (θ ⊻ σ)) \land (τ ⊻ ζ)) \land (η ⊻ ζ))$