Theorem simplbi2comtVD 45344

Description: Virtual deduction proof of simplbi2comt 503. 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 503 is simplbi2comtVD 45344 without virtual deductions and was automatically derived from simplbi2comtVD 45344.

1::	⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜑 ↔ ( 𝜓 ∧ 𝜒))   )
2:1:	⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   ((𝜓 ∧ 𝜒 ) → 𝜑)   )
3:2:	⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜓 → (𝜒 → 𝜑))   )
4:3:	⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜒 → (𝜓 → 𝜑))   )
qed:4:	⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑)))

(Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem simplbi2comtVD

Step	Hyp	Ref	Expression
1		idn1 45031	. . . . 5 ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜑 ↔ (𝜓 ∧ 𝜒))   )
2		biimpr 222	. . . . 5 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → ((𝜓 ∧ 𝜒) → 𝜑))
3	1, 2	e1a 45084	. . . 4 ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   ((𝜓 ∧ 𝜒) → 𝜑)   )
4		pm3.3 450	. . . 4 ⊢ (((𝜓 ∧ 𝜒) → 𝜑) → (𝜓 → (𝜒 → 𝜑)))
5	3, 4	e1a 45084	. . 3 ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜓 → (𝜒 → 𝜑))   )
6		pm2.04 90	. . 3 ⊢ ((𝜓 → (𝜒 → 𝜑)) → (𝜒 → (𝜓 → 𝜑)))
7	5, 6	e1a 45084	. 2 ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜒 → (𝜓 → 𝜑))   )
8	7	in1 45028	1 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-an 398  df-vd1 45027

This theorem is referenced by: (None)