Theorem spcev 2920

Description: Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Hypotheses

Ref	Expression
spcv.1	⊢ 𝐴 ∈ V
spcv.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spcev	⊢ (𝜓 → ∃𝑥𝜑)

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem spcev

Step	Hyp	Ref	Expression
1		spcv.1	. 2 ⊢ 𝐴 ∈ V
2		spcv.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	spcegv 2913	. 2 ⊢ (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by:  bnd2  4305  mss  4361  exss  4362  snnex  4589  opeldm  4979  elrnmpt1  5028  xpmlem  5203  ffoss  5667  ssimaex  5758  fvelrn  5830  funopsn  5882  eufnfv  5939  foeqcnvco  5986  cnvoprab  6460  domtr  7062  ensn1  7073  ac6sfi  7192  difinfsn  7430  0ct  7437  ctmlemr  7438  ctssdclemn0  7440  ctssdclemr  7442  ctssdc  7443  omct  7447  ctssexmid  7480  exmidfodomrlemim  7543  cc3  7624  zfz1iso  11271  fzf1o  12120  fprodntrivap  12329  nninfct  12796  ennnfonelemim  13293  ctinfom  13297  ctinf  13299  qnnen  13300  enctlem  13301  ctiunct  13309  nninfdc  13322  subctctexmid  16944  domomsubct  16945