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Theorem spcegv 2658
Description: Existential specialization, using implicit substitution. (Contributed by NM, 14-Aug-1994.)
Hypothesis
Ref Expression
spcgv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
spcegv (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem spcegv
StepHypRef Expression
1 nfcv 2194 . 2 𝑥𝐴
2 nfv 1437 . 2 𝑥𝜓
3 spcgv.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3spcegf 2653 1 (𝐴𝑉 → (𝜓 → ∃𝑥𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wex 1397  wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  spcev  2664  eqeu  2734  absneu  3470  elunii  3613  axpweq  3952  euotd  4019  brcogw  4532  opeldmg  4568  breldmg  4569  dmsnopg  4820  dff3im  5340  elunirn  5433  unielxp  5828  op1steq  5833  tfr0  5968  tfrlemibxssdm  5972  tfrlemiex  5976  ertr  6152  f1oen3g  6265  f1dom2g  6267  f1domg  6269  dom3d  6285  en1  6310  phpelm  6359  ordiso  6416  recexnq  6546  ltexprlemrl  6766  ltexprlemru  6768  recexprlemm  6780  recexprlemloc  6787  recexprlem1ssl  6789  recexprlem1ssu  6790  frecuzrdgfn  9362  climeu  10048  bj-2inf  10449
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