Theorem eqeqan12rd 2755

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 2754	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
4	3	ancoms 458	1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732

This theorem is referenced by:  fmptco  7163  axcontlem4  29000  usgredg4  29252  cusgrsize  29490  uspgr2wlkeqi  29684  clwwlkf1  30081  eigorthi  31869  goeleq12bg  35317  expdiophlem2  42979  pwssplit4  43046  fcoresf1  46984  prproropf1olem4  47380  fmtnoodd  47407  isgrim  47752