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Theorem vtoclg1f 3556
Description: Version of vtoclgf 3555 with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-10 2129 and ax-11 2146. (Contributed by BJ, 1-May-2019.)
Hypotheses
Ref Expression
vtoclg1f.nf 𝑥𝜓
vtoclg1f.maj (𝑥 = 𝐴 → (𝜑𝜓))
vtoclg1f.min 𝜑
Assertion
Ref Expression
vtoclg1f (𝐴𝑉𝜓)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem vtoclg1f
StepHypRef Expression
1 elisset 2810 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 vtoclg1f.nf . . 3 𝑥𝜓
3 vtoclg1f.min . . . 4 𝜑
4 vtoclg1f.maj . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4mpbii 232 . . 3 (𝑥 = 𝐴𝜓)
62, 5exlimi 2205 . 2 (∃𝑥 𝑥 = 𝐴𝜓)
71, 6syl 17 1 (𝐴𝑉𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wex 1773  wnf 1777  wcel 2098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-12 2166
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-clel 2805
This theorem is referenced by:  ceqsexg  3639  elabgOLD  3666  mob  3712  opeliunxp2  5843  fvopab5  7041  opeliunxp2f  8220  fprodsplit1f  15972  cnextfvval  23987  dvfsumlem2  25979  dvfsumlem2OLD  25980  dvfsumlem4  25982  bnj981  34586  dmrelrnrel  44602  fmul01  44970  fmuldfeq  44973  fmul01lt1lem1  44974  fprodcnlem  44989  stoweidlem3  45393  stoweidlem26  45416  stoweidlem31  45421  stoweidlem43  45433  stoweidlem51  45441  fourierdlem86  45582  fourierdlem89  45585  fourierdlem91  45587  salpreimagelt  46097  salpreimalegt  46099
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