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  1997  19.28  2231  2eu4  2650  r19.26-3  3093  r19.41v  3162  r3ex  3171  3reeanv  3205  r19.41  3236  rmo4  3684  rmo3f  3688  sbc3an  3801  rmo3  3835  difin2  4248  otelxp  5658  f1ounsn  7206  dfpth2  29707  dfrefrel5  38608  dfantisymrel4  38858  dfantisymrel5  38859  redvmptabs  42452  permaxsep  45099  clnbgrel  47927  grimuhgr  47986  catcinv  49499