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  4207  mss  4260  exss  4261  snnex  4484  opeldm  4870  elrnmpt1  4918  xpmlem  5091  ffoss  5539  ssimaex  5625  fvelrn  5696  eufnfv  5796  foeqcnvco  5840  cnvoprab  6301  domtr  6853  ensn1  6864  ac6sfi  6968  difinfsn  7175  0ct  7182  ctmlemr  7183  ctssdclemn0  7185  ctssdclemr  7187  ctssdc  7188  omct  7192  ctssexmid  7225  exmidfodomrlemim  7282  cc3  7353  zfz1iso  10952  fprodntrivap  11768  nninfct  12235  ennnfonelemim  12668  ctinfom  12672  ctinf  12674  qnnen  12675  enctlem  12676  ctiunct  12684  nninfdc  12697  subctctexmid  15755  domomsubct  15756