Theorem rbaibr 944

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		iba 523	. 2 ⊢ (𝜒 → (𝜓 ↔ (𝜓 ∧ 𝜒)))
2		baib.1	. 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
3	1, 2	syl6bbr 277	1 ⊢ (𝜒 → (𝜓 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383

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 196  df-an 385

This theorem is referenced by:  rbaib  945  exintrbi  1808  ssunsn2  4299  cmpfi  21021  sdrgacs  36790  nanorxor  37526  sssseq  40305