Theorem bianbi 627

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 624	. 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  628  bianassc  643  pm5.53  1006  dfifp4  1066  dfifp5  1067  an6  1447  an3andi  1484  19.28v  1996  19.28  2229  2eu4  2648  r19.26-3  3092  r19.41v  3165  r3ex  3174  3reeanv  3208  r19.41  3239  rmo4  3698  rmo3f  3702  sbc3an  3815  rmo3  3849  difin2  4260  otelxp  5675  f1ounsn  7229  dfpth2  29709  dfrefrel5  38501  dfantisymrel4  38746  dfantisymrel5  38747  redvmptabs  42341  permaxsep  44990  clnbgrel  47822  grimuhgr  47880  catcinv  49381