Theorem bianbi 628

Description: Exchanging conjunction in a biconditional. (Contributed by Peter Mazsa, 31-Jul-2023.)

Hypotheses

Ref	Expression
bianbi.1	⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
bianbi.2	⊢ (𝜓 ↔ 𝜃)

Assertion

Ref	Expression
bianbi	⊢ (𝜑 ↔ (𝜃 ∧ 𝜒))

Proof of Theorem bianbi

Step	Hyp	Ref	Expression
1		bianbi.1	. 2 ⊢ (𝜑 ↔ (𝜓 ∧ 𝜒))
2		bianbi.2	. . 3 ⊢ (𝜓 ↔ 𝜃)
3	2	anbi1i 625	. 2 ⊢ ((𝜓 ∧ 𝜒) ↔ (𝜃 ∧ 𝜒))
4	1, 3	bitri 275	1 ⊢ (𝜑 ↔ (𝜃 ∧ 𝜒))

Syntax hints:   ↔ wb 206   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  anbi12i  629  bianassc  644  pm5.53  1007  dfifp4  1067  dfifp5  1068  an6  1448  an3andi  1485  19.28v  1998  19.28  2236  2eu4  2656  r19.26-3  3098  r19.41v  3167  r3ex  3176  3reeanv  3210  r19.41  3241  rmo4  3689  rmo3f  3693  sbc3an  3806  rmo3  3840  difin2  4254  otelxp  5669  f1ounsn  7220  dfpth2  29806  dfrefrel5  38800  dfdisjALTV5a  39006  dfantisymrel4  39067  dfantisymrel5  39068  petseq  39179  redvmptabs  42682  permaxsep  45315  clnbgrel  48141  grimuhgr  48200  catcinv  49711