Theorem pm4.71r 388

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		pm4.71 387	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓)))
2		ancom 264	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
3	2	bibi2i 226	. 2 ⊢ ((𝜑 ↔ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜑)))
4	1, 3	bitri 183	1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜓 ∧ 𝜑)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm4.71ri  390  pm4.71rd  392