Theorem spcev 2820

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 2813	. 2 ⊢ (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2725

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727

This theorem is referenced by:  bnd2  4151  mss  4203  exss  4204  snnex  4425  opeldm  4806  elrnmpt1  4854  xpmlem  5023  ffoss  5463  ssimaex  5546  fvelrn  5615  eufnfv  5714  foeqcnvco  5757  cnvoprab  6198  domtr  6747  ensn1  6758  ac6sfi  6860  difinfsn  7061  0ct  7068  ctmlemr  7069  ctssdclemn0  7071  ctssdclemr  7073  ctssdc  7074  omct  7078  ctssexmid  7110  exmidfodomrlemim  7153  cc3  7205  zfz1iso  10750  fprodntrivap  11521  ennnfonelemim  12353  ctinfom  12357  ctinf  12359  qnnen  12360  enctlem  12361  ctiunct  12369  nninfdc  12382  subctctexmid  13841