Theorem spcev 2872

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 2865	. 2 ⊢ (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775

This theorem is referenced by:  bnd2  4225  mss  4278  exss  4279  snnex  4503  opeldm  4890  elrnmpt1  4938  xpmlem  5112  ffoss  5566  ssimaex  5653  fvelrn  5724  funopsn  5775  eufnfv  5828  foeqcnvco  5872  cnvoprab  6333  domtr  6890  ensn1  6901  ac6sfi  7010  difinfsn  7217  0ct  7224  ctmlemr  7225  ctssdclemn0  7227  ctssdclemr  7229  ctssdc  7230  omct  7234  ctssexmid  7267  exmidfodomrlemim  7325  cc3  7400  zfz1iso  11008  fprodntrivap  11970  nninfct  12437  ennnfonelemim  12870  ctinfom  12874  ctinf  12876  qnnen  12877  enctlem  12878  ctiunct  12886  nninfdc  12899  subctctexmid  16078  domomsubct  16079