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Theorem vtocldf 2661
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 113 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
63, 5alrimi 1456 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
7 vtocld.3 . . 3 (𝜑𝜓)
83, 7alrimi 1456 . 2 (𝜑 → ∀𝑥𝜓)
9 vtocld.1 . 2 (𝜑𝐴𝑉)
10 vtoclgft 2660 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜒) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) ∧ ∀𝑥𝜓) ∧ 𝐴𝑉) → 𝜒)
111, 2, 6, 8, 9, 10syl221anc 1181 1 (𝜑𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wnf 1390  wcel 1434  wnfc 2210
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614
This theorem is referenced by:  vtocld  2662  peano2  4373  iota2df  4958
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