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Theorem vtoclegft 2691
Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef 2692.) (Contributed by NM, 7-Nov-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)
Assertion
Ref Expression
vtoclegft ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem vtoclegft
StepHypRef Expression
1 elisset 2633 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
2 exim 1535 . . . 4 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥𝜑))
31, 2mpan9 275 . . 3 ((𝐴𝐵 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → ∃𝑥𝜑)
433adant2 962 . 2 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → ∃𝑥𝜑)
5 19.9t 1578 . . 3 (Ⅎ𝑥𝜑 → (∃𝑥𝜑𝜑))
653ad2ant2 965 . 2 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → (∃𝑥𝜑𝜑))
74, 6mpbid 145 1 ((𝐴𝐵 ∧ Ⅎ𝑥𝜑 ∧ ∀𝑥(𝑥 = 𝐴𝜑)) → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  w3a 924  wal 1287   = wceq 1289  wnf 1394  wex 1426  wcel 1438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by: (None)
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