Theorem bianbi 626

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 623	. 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  627  bianassc  642  pm5.53  1005  dfifp4  1067  dfifp5  1068  an3andi  1482  19.28v  1990  19.28  2224  2eu4  2652  r19.26-3  3114  r19.41v  3191  r3ex  3200  3reeanv  3231  r19.41  3264  rmo4  3746  rmo3f  3750  sbc3an  3868  rmo3  3905  difin2  4315  otelxp  5743  dfrefrel5  38421  dfantisymrel4  38665  dfantisymrel5  38666  clnbgrel  47633  grimuhgr  47691