Theorem anbi1ci 638

Description: Variant of anbi1i 636 with commutation. (Contributed by Peter Mazsa, 7-Mar-2020.)

Hypothesis

Ref	Expression
anbi.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
anbi1ci	⊢ ((𝜒 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒))

Proof of Theorem anbi1ci

Step	Hyp	Ref	Expression
1		anbi.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
2	1	anbi2i 635	. 2 ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜒 ∧ 𝜓))
3	2	biancomi 468	1 ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜓 ∧ 𝜒))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  dfid3  5548  imai  6071  frpoind  6345  dfac5lem3  10208  cf0  10332  eqcuts2  28178  coep  36610  brtxp  36735  sscoid  36768  brapply  36793  dfrdg4  36808  wl-df4-3mintru2  38504  rnxrncnvepres  39449  rnxrnidres  39450  pmapglb  40921  polval2N  41057  rp-fakeoranass  44609  alephiso2  44653