Theorem spcev 2901

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 2894	. 2 ⊢ (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804

This theorem is referenced by:  bnd2  4263  mss  4318  exss  4319  snnex  4545  opeldm  4934  elrnmpt1  4983  xpmlem  5157  ffoss  5616  ssimaex  5707  fvelrn  5778  funopsn  5829  eufnfv  5884  foeqcnvco  5930  cnvoprab  6398  domtr  6958  ensn1  6969  ac6sfi  7086  difinfsn  7298  0ct  7305  ctmlemr  7306  ctssdclemn0  7308  ctssdclemr  7310  ctssdc  7311  omct  7315  ctssexmid  7348  exmidfodomrlemim  7411  cc3  7486  zfz1iso  11104  fprodntrivap  12144  nninfct  12611  ennnfonelemim  13044  ctinfom  13048  ctinf  13050  qnnen  13051  enctlem  13052  ctiunct  13060  nninfdc  13073  subctctexmid  16601  domomsubct  16602