Theorem eqeqan12rd 2097

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 2096	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
4	3	ancoms 264	1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-4 1440  ax-17 1459  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074

This theorem is referenced by: (None)