Theorem pm4.71r 394

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		pm4.71 393	. 2 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜑 ∧ 𝜓)))
2		ancom 266	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
3	2	bibi2i 227	. 2 ⊢ ((𝜑 ↔ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ (𝜓 ∧ 𝜑)))
4	1, 3	bitri 184	1 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 ↔ (𝜓 ∧ 𝜑)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  pm4.71ri  396  pm4.71rd  398  iotaexel  6043  iftrueb01  7583