Theorem spcev 2856

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 2849	. 2 ⊢ (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
4	1, 3	ax-mp 5	1 ⊢ (𝜓 → ∃𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  ∃wex 1503   ∈ wcel 2164  Vcvv 2760

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762

This theorem is referenced by:  bnd2  4203  mss  4256  exss  4257  snnex  4480  opeldm  4866  elrnmpt1  4914  xpmlem  5087  ffoss  5533  ssimaex  5619  fvelrn  5690  eufnfv  5790  foeqcnvco  5834  cnvoprab  6289  domtr  6841  ensn1  6852  ac6sfi  6956  difinfsn  7161  0ct  7168  ctmlemr  7169  ctssdclemn0  7171  ctssdclemr  7173  ctssdc  7174  omct  7178  ctssexmid  7211  exmidfodomrlemim  7263  cc3  7330  zfz1iso  10915  fprodntrivap  11730  nninfct  12181  ennnfonelemim  12584  ctinfom  12588  ctinf  12590  qnnen  12591  enctlem  12592  ctiunct  12600  nninfdc  12613  subctctexmid  15561