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  5740  eqfnfv2  5741  f1mpt  5907  xpopth  6334  f1o2ndf1  6388  ecopoveq  6794  xpdom2  7010  djune  7271  addpipqqs  7583  enq0enq  7644  enq0sym  7645  enq0tr  7647  enq0breq  7649  preqlu  7685  cnegexlem1  8347  neg11  8423  subeqrev  8548  cnref1o  9878  xneg11  10062  modlteq  10652  sq11  10867  qsqeqor  10905  fz1eqb  11045  eqwrd  11147  s111  11201  ccatopth  11290  wrd2ind  11297  cj11  11459  sqrt11  11593  sqabs  11636  recan  11663  reeff1  12254  efieq  12289  xpsff1o  13425  ismhm  13537  isdomn  14276  tgtop11  14793  ioocosf1o  15571  mpodvdsmulf1o  15707  iswlk  16134