Theorem bianir 1074

Description: A closed form of mpbir 234, analogous to pm2.27 43 (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 223	. 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜑 → 𝜓))
2	1	impcom 413	1 ⊢ ((𝜑 ∧ (𝜓 ↔ 𝜑)) → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  fiun  7942  suppimacnv  8172  lgsqrmodndvds  27548  bnj970  35376  bnj1001  35388  bj-bibibi  37212  bj-axreprepsep  37745