Theorem rbaibr 541

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 466	. 2 ⊢ (𝜑 ↔ (𝜒 ∧ 𝜓))
3	2	baibr 540	1 ⊢ (𝜒 → (𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399

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 210  df-an 400

This theorem is referenced by:  rbaib  542  exintrbi  1899  moeuex  2581  sssseq  3905  ssunsn2  4726  sdrgacs  19799  cmpfi  22259  nanorxor  41537