Theorem pm5.501 244

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 173	. 2 ⊢ (𝜑 → (𝜓 → (𝜑 ↔ 𝜓)))
2		biimp 118	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
3	2	com12 30	. 2 ⊢ (𝜑 → ((𝜑 ↔ 𝜓) → 𝜓))
4	1, 3	impbid 129	1 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  ibib  245  ibibr  246  pm5.1  601  pm5.18dc  884  biassdc  1406