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Theorem vtoclg1f 3564
Description: Version of vtoclgf 3563 with one non-freeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-11 2151 and ax-13 2381. (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 3503 . 2 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 vtoclg1f.nf . . 3 𝑥𝜓
3 vtoclg1f.min . . . 4 𝜑
4 vtoclg1f.maj . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
53, 4mpbii 234 . . 3 (𝑥 = 𝐴𝜓)
62, 5exlimi 2207 . 2 (∃𝑥 𝑥 = 𝐴𝜓)
71, 6syl 17 1 (𝐴𝑉𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wex 1771  wnf 1775  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776  df-cleq 2811  df-clel 2890
This theorem is referenced by:  vtoclg  3565  ceqsexg  3643  elabg  3663  mob  3705  opeliunxp2  5702  fvopab5  6792  opeliunxp2f  7865  fprodsplit1f  15332  cnextfvval  22601  dvfsumlem2  24551  dvfsumlem4  24553  bnj981  32121  dmrelrnrel  41366  fmul01  41737  fmuldfeq  41740  fmul01lt1lem1  41741  stoweidlem3  42165  stoweidlem26  42188  stoweidlem31  42193  stoweidlem43  42205  stoweidlem51  42213  fourierdlem86  42354  fourierdlem89  42357  fourierdlem91  42359  salpreimagelt  42863  salpreimalegt  42865
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