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  4310  mss  4366  exss  4367  snnex  4594  opeldm  4984  elrnmpt1  5033  xpmlem  5208  ffoss  5672  ssimaex  5764  fvelrn  5839  funopsn  5891  eufnfv  5949  foeqcnvco  5996  cnvoprab  6470  domtr  7072  ensn1  7083  ac6sfi  7202  difinfsn  7440  0ct  7447  ctmlemr  7448  ctssdclemn0  7450  ctssdclemr  7452  ctssdc  7453  omct  7457  ctssexmid  7490  exmidfodomrlemim  7553  cc3  7634  zfz1iso  11293  fzf1o  12142  fprodntrivap  12351  nninfct  12818  ennnfonelemim  13315  ctinfom  13319  ctinf  13321  qnnen  13322  enctlem  13323  ctiunct  13331  nninfdc  13344  subctctexmid  17030  domomsubct  17031