Theorem spcev 2898

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 2891	. 2 ⊢ (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  bnd2  4257  mss  4312  exss  4313  snnex  4539  opeldm  4926  elrnmpt1  4975  xpmlem  5149  ffoss  5606  ssimaex  5697  fvelrn  5768  funopsn  5819  eufnfv  5874  foeqcnvco  5920  cnvoprab  6386  domtr  6945  ensn1  6956  ac6sfi  7068  difinfsn  7278  0ct  7285  ctmlemr  7286  ctssdclemn0  7288  ctssdclemr  7290  ctssdc  7291  omct  7295  ctssexmid  7328  exmidfodomrlemim  7390  cc3  7465  zfz1iso  11076  fprodntrivap  12110  nninfct  12577  ennnfonelemim  13010  ctinfom  13014  ctinf  13016  qnnen  13017  enctlem  13018  ctiunct  13026  nninfdc  13039  subctctexmid  16425  domomsubct  16426