Theorem eqeqan12rd 2187

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 2186	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
4	3	ancoms 266	1 ⊢ ((𝜓 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163

This theorem is referenced by:  omp1eomlem  7071