Theorem eqeqan12rd 2053

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 2052	. 2 ⊢ ((φ ∧ ψ) → (A = 𝐶 ↔ B = 𝐷))
4	3	ancoms 255	1 ⊢ ((ψ ∧ φ) → (A = 𝐶 ↔ B = 𝐷))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-cleq 2030

This theorem is referenced by: (None)