Theorem eqeqan12rd 2775

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.)

Hypotheses

Ref	Expression
eqeqan12rd.1	⊢ (𝜑 → 𝐴 = 𝐵)
eqeqan12rd.2	⊢ (𝜓 → 𝐶 = 𝐷)

Assertion

Ref	Expression
eqeqan12rd	⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Proof of Theorem eqeqan12rd

Step	Hyp	Ref	Expression
1		eqeqan12rd.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		eqeqan12rd.2	. . 3 ⊢ (𝜓 → 𝐶 = 𝐷)
3	1, 2	eqeqan12d 2774	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
4	3	ancoms 464	1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752

This theorem is used by:  fmptco  7130  axcontlem4  29545  usgredg4  29798  cusgrsize  30035  uspgr2wlkeqi  30228  clwwlkf1  30640  eigorthi  32439  goeleq12bg  36114  expdiophlem2  44029  pwssplit4  44091  fcoresf1  48139  prproropf1olem4  48588  fmtnoodd  48618  isgrim  48980