Theorem pm5.501 367

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

Assertion

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

Proof of Theorem pm5.501

Step	Hyp	Ref	Expression
1		pm5.1im 262	. 2 ⊢ (𝜑 → (𝜓 → (𝜑 ↔ 𝜓)))
2		biimp 214	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
3	2	com12 32	. 2 ⊢ (𝜑 → ((𝜑 ↔ 𝜓) → 𝜓))
4	1, 3	impbid 211	1 ⊢ (𝜑 → (𝜓 ↔ (𝜑 ↔ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205

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 206

This theorem is referenced by:  ibib  368  ibibr  369  nbn2  371  pm5.18  383  biass  386  pm5.1  821  vn0  4272  sadadd2lem2  16157  isclo  22238  nrmmetd  23730  bj-bibibi  34768