Theorem simplbi2comtVD 41229

Description: Virtual deduction proof of simplbi2comt 504. 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 504 is simplbi2comtVD 41229 without virtual deductions and was automatically derived from simplbi2comtVD 41229.

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 40915	. . . . 5 ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜑 ↔ (𝜓 ∧ 𝜒))   )
2		biimpr 222	. . . . 5 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → ((𝜓 ∧ 𝜒) → 𝜑))
3	1, 2	e1a 40968	. . . 4 ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   ((𝜓 ∧ 𝜒) → 𝜑)   )
4		pm3.3 451	. . . 4 ⊢ (((𝜓 ∧ 𝜒) → 𝜑) → (𝜓 → (𝜒 → 𝜑)))
5	3, 4	e1a 40968	. . 3 ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜓 → (𝜒 → 𝜑))   )
6		pm2.04 90	. . 3 ⊢ ((𝜓 → (𝜒 → 𝜑)) → (𝜒 → (𝜓 → 𝜑)))
7	5, 6	e1a 40968	. 2 ⊢ (   (𝜑 ↔ (𝜓 ∧ 𝜒))   ▶   (𝜒 → (𝜓 → 𝜑))   )
8	7	in1 40912	1 ⊢ ((𝜑 ↔ (𝜓 ∧ 𝜒)) → (𝜒 → (𝜓 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398

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 399  df-vd1 40911

This theorem is referenced by: (None)