Theorem pm4.71r 612

Description: Implication in terms of biconditional and conjunction. Theorem *4.71 of [WhiteheadRussell] p. 120 (with conjunct reversed). (Contributed by NM, 25-Jul-1999.)

Assertion

Ref	Expression
pm4.71r	⊢ ((φ → ψ) ↔ (φ ↔ (ψ ∧ φ)))

Proof of Theorem pm4.71r

Step	Hyp	Ref	Expression
1		pm4.71 611	. 2 ⊢ ((φ → ψ) ↔ (φ ↔ (φ ∧ ψ)))
2		ancom 437	. . 3 ⊢ ((φ ∧ ψ) ↔ (ψ ∧ φ))
3	2	bibi2i 304	. 2 ⊢ ((φ ↔ (φ ∧ ψ)) ↔ (φ ↔ (ψ ∧ φ)))
4	1, 3	bitri 240	1 ⊢ ((φ → ψ) ↔ (φ ↔ (ψ ∧ 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:  pm4.71ri  614  pm4.71rd  616