Theorem spcev 2902

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 2895	. 2 ⊢ (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805

This theorem is referenced by:  bnd2  4269  mss  4324  exss  4325  snnex  4551  opeldm  4940  elrnmpt1  4989  xpmlem  5164  ffoss  5625  ssimaex  5716  fvelrn  5786  funopsn  5838  eufnfv  5895  foeqcnvco  5941  cnvoprab  6408  domtr  7002  ensn1  7013  ac6sfi  7130  difinfsn  7342  0ct  7349  ctmlemr  7350  ctssdclemn0  7352  ctssdclemr  7354  ctssdc  7355  omct  7359  ctssexmid  7392  exmidfodomrlemim  7455  cc3  7530  zfz1iso  11151  fzf1o  11999  fprodntrivap  12208  nninfct  12675  ennnfonelemim  13108  ctinfom  13112  ctinf  13114  qnnen  13115  enctlem  13116  ctiunct  13124  nninfdc  13137  subctctexmid  16705  domomsubct  16706