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Theorem vtocldf 2599
 Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
vtocld.1 (φA 𝑉)
vtocld.2 ((φ x = 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 ((φ x = A) → (ψχ))
54ex 108 . . 3 (φ → (x = A → (ψχ)))
63, 5alrimi 1412 . 2 (φx(x = A → (ψχ)))
7 vtocld.3 . . 3 (φψ)
83, 7alrimi 1412 . 2 (φxψ)
9 vtocld.1 . 2 (φA 𝑉)
10 vtoclgft 2598 . 2 (((xA xχ) (x(x = A → (ψχ)) xψ) A 𝑉) → χ)
111, 2, 6, 8, 9, 10syl221anc 1145 1 (φχ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  Ⅎwnf 1346   ∈ wcel 1390  Ⅎwnfc 2162 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553 This theorem is referenced by:  vtocld  2600  peano2  4261  iota2df  4834
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