Theorem eqeqan12d 2245

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 2242	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
4	1, 2, 3	syl2an 289	1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222

This theorem is referenced by:  eqeqan12rd  2246  eqfnfv  5737  eqfnfv2  5738  f1mpt  5904  xpopth  6331  f1o2ndf1  6385  ecopoveq  6790  xpdom2  7003  djune  7261  addpipqqs  7573  enq0enq  7634  enq0sym  7635  enq0tr  7637  enq0breq  7639  preqlu  7675  cnegexlem1  8337  neg11  8413  subeqrev  8538  cnref1o  9863  xneg11  10047  modlteq  10636  sq11  10851  qsqeqor  10889  fz1eqb  11029  eqwrd  11130  s111  11184  ccatopth  11269  wrd2ind  11276  cj11  11437  sqrt11  11571  sqabs  11614  recan  11641  reeff1  12232  efieq  12267  xpsff1o  13403  ismhm  13515  isdomn  14254  tgtop11  14771  ioocosf1o  15549  mpodvdsmulf1o  15685  iswlk  16095