Theorem pm5.501 243

Description: Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 24-Jan-2013.)

Assertion

Ref	Expression
pm5.501	⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓)))

Proof of Theorem pm5.501

Step	Hyp	Ref	Expression
1		pm5.1im 172	. 2 ⊢ (𝜑 → (𝜓 → (𝜑 ↔ 𝜓)))
2		biimp 117	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
3	2	com12 30	. 2 ⊢ (𝜑 → ((𝜑 ↔ 𝜓) → 𝜓))
4	1, 3	impbid 128	1 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓)))

Syntax hints:   → wi 4   ↔ 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:  ibib  244  ibibr  245  pm5.1  591  pm5.18dc  873  biassdc  1385