Theorem eqeqan12rd 2369

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 2368	. 2 ⊢ ((φ ∧ ψ) → (A = C ↔ B = D))
4	3	ancoms 439	1 ⊢ ((ψ ∧ φ) → (A = C ↔ B = D))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346

This theorem is used by: (None)