Theorem bimsc1 904

Description: Removal of conjunct from one side of an equivalence. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
bimsc1	⊢ (((φ → ψ) ∧ (χ ↔ (ψ ∧ φ))) → (χ ↔ φ))

Proof of Theorem bimsc1

Step	Hyp	Ref	Expression
1		simpr 447	. . . 4 ⊢ ((ψ ∧ φ) → φ)
2		ancr 532	. . . 4 ⊢ ((φ → ψ) → (φ → (ψ ∧ φ)))
3	1, 2	impbid2 195	. . 3 ⊢ ((φ → ψ) → ((ψ ∧ φ) ↔ φ))
4	3	bibi2d 309	. 2 ⊢ ((φ → ψ) → ((χ ↔ (ψ ∧ φ)) ↔ (χ ↔ φ)))
5	4	biimpa 470	1 ⊢ (((φ → ψ) ∧ (χ ↔ (ψ ∧ φ))) → (χ ↔ φ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358

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 177  df-an 360

This theorem is referenced by: (None)