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Theorem vtocldf 2737
 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 1502 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
7 vtocld.3 . . 3 (𝜑𝜓)
83, 7alrimi 1502 . 2 (𝜑 → ∀𝑥𝜓)
9 vtocld.1 . 2 (𝜑𝐴𝑉)
10 vtoclgft 2736 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜒) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) ∧ ∀𝑥𝜓) ∧ 𝐴𝑉) → 𝜒)
111, 2, 6, 8, 9, 10syl221anc 1227 1 (𝜑𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   = wceq 1331  Ⅎwnf 1436   ∈ wcel 1480  Ⅎwnfc 2268 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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688 This theorem is referenced by:  vtocld  2738  peano2  4509  iota2df  5112
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