Theorem bianir 1056

Description: A closed form of mpbir 230, analogous to pm2.27 42 (assertion). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Roger Witte, 17-Aug-2020.)

Assertion

Ref	Expression
bianir	⊢ ((𝜑 ∧ (𝜓 ↔ 𝜑)) → 𝜓)

Proof of Theorem bianir

Step	Hyp	Ref	Expression
1		biimpr 219	. 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜑 → 𝜓))
2	1	impcom 408	1 ⊢ ((𝜑 ∧ (𝜓 ↔ 𝜑)) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396

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  df-an 397

This theorem is referenced by:  fiun  7785  suppimacnv  7990  lgsqrmodndvds  26501  bnj970  32927  bnj1001  32939  bj-bibibi  34768  isomuspgrlem2b  45281