Theorem eqeqan12d 2100

Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
eqeqan12d.1	⊢ (𝜑 → 𝐴 = 𝐵)
eqeqan12d.2	⊢ (𝜓 → 𝐶 = 𝐷)

Assertion

Ref	Expression
eqeqan12d	⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Proof of Theorem eqeqan12d

Step	Hyp	Ref	Expression
1		eqeqan12d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		eqeqan12d.2	. 2 ⊢ (𝜓 → 𝐶 = 𝐷)
3		eqeq12 2097	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
4	1, 2, 3	syl2an 283	1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1287

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 1379  ax-gen 1381  ax-4 1443  ax-17 1462  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-cleq 2078

This theorem is referenced by:  eqeqan12rd  2101  eqfnfv  5353  eqfnfv2  5354  f1mpt  5504  xpopth  5896  f1o2ndf1  5943  ecopoveq  6332  xpdom2  6492  djune  6705  addpipqqs  6865  enq0enq  6926  enq0sym  6927  enq0tr  6929  enq0breq  6931  preqlu  6967  cnegexlem1  7593  neg11  7669  subeqrev  7790  cnref1o  9057  xneg11  9220  modlteq  9724  sq11  9917  fz1eqb  10087  cj11  10226  sqrt11  10359  sqabs  10402  recan  10429