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  4256  mss  4311  exss  4312  snnex  4538  opeldm  4925  elrnmpt1  4974  xpmlem  5148  ffoss  5603  ssimaex  5694  fvelrn  5765  funopsn  5816  eufnfv  5869  foeqcnvco  5913  cnvoprab  6378  domtr  6935  ensn1  6946  ac6sfi  7056  difinfsn  7263  0ct  7270  ctmlemr  7271  ctssdclemn0  7273  ctssdclemr  7275  ctssdc  7276  omct  7280  ctssexmid  7313  exmidfodomrlemim  7375  cc3  7450  zfz1iso  11058  fprodntrivap  12090  nninfct  12557  ennnfonelemim  12990  ctinfom  12994  ctinf  12996  qnnen  12997  enctlem  12998  ctiunct  13006  nninfdc  13019  subctctexmid  16325  domomsubct  16326