Theorem simplbi2comtVD 39873

Description: Virtual deduction proof of simplbi2comt 496. 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 496 is simplbi2comtVD 39873 without virtual deductions and was automatically derived from simplbi2comtVD 39873.

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 39549	. . . . 5 ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜑 ↔ (𝜓 ∧ 𝜒))   )
2		biimpr 212	. . . . 5 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → ((𝜓 ∧ 𝜒) → 𝜑))
3	1, 2	e1a 39611	. . . 4 ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   ((𝜓 ∧ 𝜒) → 𝜑)   )
4		pm3.3 440	. . . 4 ⊢ (((𝜓 ∧ 𝜒) → 𝜑) → (𝜓 → (𝜒 → 𝜑)))
5	3, 4	e1a 39611	. . 3 ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜓 → (𝜒 → 𝜑))   )
6		pm2.04 90	. . 3 ⊢ ((𝜓 → (𝜒 → 𝜑)) → (𝜒 → (𝜓 → 𝜑)))
7	5, 6	e1a 39611	. 2 ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜒 → (𝜓 → 𝜑))   )
8	7	in1 39546	1 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385

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 199  df-an 386  df-vd1 39545

This theorem is referenced by: (None)