Theorem spcev 2899

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 2892	. 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 2800

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 2802

This theorem is referenced by:  bnd2  4261  mss  4316  exss  4317  snnex  4543  opeldm  4932  elrnmpt1  4981  xpmlem  5155  ffoss  5612  ssimaex  5703  fvelrn  5774  funopsn  5825  eufnfv  5880  foeqcnvco  5926  cnvoprab  6394  domtr  6954  ensn1  6965  ac6sfi  7080  difinfsn  7290  0ct  7297  ctmlemr  7298  ctssdclemn0  7300  ctssdclemr  7302  ctssdc  7303  omct  7307  ctssexmid  7340  exmidfodomrlemim  7402  cc3  7477  zfz1iso  11095  fprodntrivap  12135  nninfct  12602  ennnfonelemim  13035  ctinfom  13039  ctinf  13041  qnnen  13042  enctlem  13043  ctiunct  13051  nninfdc  13064  subctctexmid  16537  domomsubct  16538