Theorem spcev 2859

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 2852	. 2 ⊢ (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167  Vcvv 2763

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  bnd2  4206  mss  4259  exss  4260  snnex  4483  opeldm  4869  elrnmpt1  4917  xpmlem  5090  ffoss  5536  ssimaex  5622  fvelrn  5693  eufnfv  5793  foeqcnvco  5837  cnvoprab  6292  domtr  6844  ensn1  6855  ac6sfi  6959  difinfsn  7166  0ct  7173  ctmlemr  7174  ctssdclemn0  7176  ctssdclemr  7178  ctssdc  7179  omct  7183  ctssexmid  7216  exmidfodomrlemim  7268  cc3  7335  zfz1iso  10933  fprodntrivap  11749  nninfct  12208  ennnfonelemim  12641  ctinfom  12645  ctinf  12647  qnnen  12648  enctlem  12649  ctiunct  12657  nninfdc  12670  subctctexmid  15645