Metamath Proof Explorer

Theorem simplbi2comtVD

Description: Virtual deduction proof of simplbi2comt . 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. simplbi2comt is simplbi2comtVD without virtual deductions and was automatically derived from simplbi2comtVD .

1:: |- (. ( ph <-> ( ps /\ ch ) ) ->. ( ph <-> ( ps /\ ch ) ) ).
2:1: |- (. ( ph <-> ( ps /\ ch ) ) ->. ( ( ps /\ ch ) -> ph ) ).
3:2: |- (. ( ph <-> ( ps /\ ch ) ) ->. ( ps -> ( ch -> ph ) ) ).
4:3: |- (. ( ph <-> ( ps /\ ch ) ) ->. ( ch -> ( ps -> ph ) ) ).
qed:4: |- ( ( ph <-> ( ps /\ ch ) ) -> ( ch -> ( ps -> ph ) ) )
(Contributed by Alan Sare, 22-Jul-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion simplbi2comtVD ( ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) → ( 𝜒 → ( 𝜓 → 𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 idn1 ⊢ (    ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) )    ▶    ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) )    )
2 biimpr ⊢ ( ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) → ( ( 𝜓 ∧ 𝜒 ) → 𝜑 ) )
3 1 2 e1a ⊢ (    ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) )    ▶    ( ( 𝜓 ∧ 𝜒 ) → 𝜑 )    )
4 pm3.3 ⊢ ( ( ( 𝜓 ∧ 𝜒 ) → 𝜑 ) → ( 𝜓 → ( 𝜒 → 𝜑 ) ) )
5 3 4 e1a ⊢ (    ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) )    ▶    ( 𝜓 → ( 𝜒 → 𝜑 ) )    )
6 pm2.04 ⊢ ( ( 𝜓 → ( 𝜒 → 𝜑 ) ) → ( 𝜒 → ( 𝜓 → 𝜑 ) ) )
7 5 6 e1a ⊢ (    ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) )    ▶    ( 𝜒 → ( 𝜓 → 𝜑 ) )    )
8 7 in1 ⊢ ( ( 𝜑 ↔ ( 𝜓 ∧ 𝜒 ) ) → ( 𝜒 → ( 𝜓 → 𝜑 ) ) )