Theorem rbaibr 542

Description: Move conjunction outside of biconditional. (Contributed by Mario Carneiro, 11-Sep-2015.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)

Hypothesis

Ref	Expression
baib.1	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))

Assertion

Ref	Expression
rbaibr	⊢ (𝜒 → (𝜓 ↔ 𝜑))

Proof of Theorem rbaibr

Step	Hyp	Ref	Expression
1		baib.1	. . 3 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2	1	biancomi 463	. 2 ⊢ (𝜑 ↔ (𝜒 ∧ 𝜓))
3	2	baibr 541	1 ⊢ (𝜒 → (𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 207   ∧ 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 208  df-an 397

This theorem is referenced by:  rbaib  543  exintrbi  1898  moeuex  2586  sssseq  3933  ssunsn2  4758  sdrgacs  20773  cmpfi  23391  fimgmcyc  43020  nanorxor  44749