Theorem spcev 2920

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 2913	. 2 ⊢ (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 → ∃𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is used by:  bnd2  4312  mss  4368  exss  4369  snnex  4596  opeldm  4986  elrnmpt1  5035  xpmlem  5211  ffoss  5675  ssimaex  5767  fvelrn  5842  funopsn  5894  eufnfv  5952  foeqcnvco  5999  cnvoprab  6474  domtr  7076  ensn1  7087  ac6sfi  7206  difinfsn  7446  0ct  7453  ctmlemr  7454  ctssdclemn0  7456  ctssdclemr  7458  ctssdc  7459  omct  7463  ctssexmid  7496  exmidfodomrlemim  7559  cc3  7640  zfz1iso  11315  fzf1o  12167  fprodntrivap  12376  nninfct  12843  ennnfonelemim  13373  ctinfom  13377  ctinf  13379  qnnen  13380  enctlem  13381  ctiunct  13389  nninfdc  13402  subctctexmid  17235  domomsubct  17236