Theorem eqeqan12d 2254

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 2251	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))
4	1, 2, 3	syl2an 289	1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-4 1563  ax-17 1579  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231

This theorem is used by:  eqeqan12rd  2255  eqfnfv  5806  eqfnfv2  5807  f1mpt  5977  xpopth  6410  f1o2ndf1  6464  ecopoveq  6904  xpdom2  7129  djune  7419  addpipqqs  7738  enq0enq  7799  enq0sym  7800  enq0tr  7802  enq0breq  7804  preqlu  7840  cnegexlem1  8503  neg11  8579  subeqrev  8704  cnref1o  10062  xneg11  10247  modlteq  10848  sq11  11063  qsqeqor  11101  fz1eqb  11244  eqwrd  11360  s111  11414  ccatopth  11503  wrd2ind  11510  cj11  11686  sqrt11  11820  sqabs  11864  recan  11891  reeff1  12485  efieq  12520  xpsff1o  13721  ismhm  13819  isdomn  14629  tgtop11  15229  ioocosf1o  16008  mpodvdsmulf1o  16206  iswlk  16686