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  5725  eqfnfv2  5726  f1mpt  5888  xpopth  6312  f1o2ndf1  6364  ecopoveq  6767  xpdom2  6978  djune  7233  addpipqqs  7545  enq0enq  7606  enq0sym  7607  enq0tr  7609  enq0breq  7611  preqlu  7647  cnegexlem1  8309  neg11  8385  subeqrev  8510  cnref1o  9834  xneg11  10018  modlteq  10606  sq11  10821  qsqeqor  10859  fz1eqb  10999  eqwrd  11098  s111  11150  ccatopth  11234  wrd2ind  11241  cj11  11402  sqrt11  11536  sqabs  11579  recan  11606  reeff1  12197  efieq  12232  xpsff1o  13368  ismhm  13480  isdomn  14218  tgtop11  14735  ioocosf1o  15513  mpodvdsmulf1o  15649