Theorem spcev 2807

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 2800	. 2 ⊢ (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1335  ∃wex 1472   ∈ wcel 2128  Vcvv 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714

This theorem is referenced by:  bnd2  4133  mss  4185  exss  4186  snnex  4406  opeldm  4786  elrnmpt1  4834  xpmlem  5003  ffoss  5443  ssimaex  5526  fvelrn  5595  eufnfv  5692  foeqcnvco  5735  cnvoprab  6175  domtr  6723  ensn1  6734  ac6sfi  6836  difinfsn  7034  0ct  7041  ctmlemr  7042  ctssdclemn0  7044  ctssdclemr  7046  ctssdc  7047  omct  7051  ctssexmid  7076  exmidfodomrlemim  7119  cc3  7171  zfz1iso  10694  fprodntrivap  11463  ennnfonelemim  12125  ctinfom  12129  ctinf  12131  qnnen  12132  enctlem  12133  ctiunct  12141  subctctexmid  13533