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  5556  imai  6090  frpoind  6371  dfac5lem3  10243  cf0  10367  eqcuts2  28229  coep  36732  brtxp  36857  sscoid  36890  brapply  36915  dfrdg4  36930  wl-df4-3mintru2  38678  rnxrncnvepres  39621  rnxrnidres  39622  pmapglb  41093  polval2N  41229  rp-fakeoranass  44776  alephiso2  44820