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 1570

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2752

This theorem is used by:  fmptco  7118  axcontlem4  29478  usgredg4  29731  cusgrsize  29968  uspgr2wlkeqi  30161  clwwlkf1  30573  eigorthi  32372  goeleq12bg  36035  expdiophlem2  43967  pwssplit4  44034  fcoresf1  48061  prproropf1olem4  48510  fmtnoodd  48540  isgrim  48902