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Theorem vtocldf 2711
Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
vtocld.1 (𝜑𝐴𝑉)
vtocld.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
vtocld.3 (𝜑𝜓)
vtocldf.4 𝑥𝜑
vtocldf.5 (𝜑𝑥𝐴)
vtocldf.6 (𝜑 → Ⅎ𝑥𝜒)
Assertion
Ref Expression
vtocldf (𝜑𝜒)

Proof of Theorem vtocldf
StepHypRef Expression
1 vtocldf.5 . 2 (𝜑𝑥𝐴)
2 vtocldf.6 . 2 (𝜑 → Ⅎ𝑥𝜒)
3 vtocldf.4 . . 3 𝑥𝜑
4 vtocld.2 . . . 4 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
54ex 114 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
63, 5alrimi 1487 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
7 vtocld.3 . . 3 (𝜑𝜓)
83, 7alrimi 1487 . 2 (𝜑 → ∀𝑥𝜓)
9 vtocld.1 . 2 (𝜑𝐴𝑉)
10 vtoclgft 2710 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜒) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) ∧ ∀𝑥𝜓) ∧ 𝐴𝑉) → 𝜒)
111, 2, 6, 8, 9, 10syl221anc 1212 1 (𝜑𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1314   = wceq 1316  wnf 1421  wcel 1465  wnfc 2245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by:  vtocld  2712  peano2  4479  iota2df  5082
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