MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtocldf Structured version   Visualization version   GIF version

Theorem vtocldf 3554
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 415 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
63, 5alrimi 2206 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
7 vtocld.3 . . 3 (𝜑𝜓)
83, 7alrimi 2206 . 2 (𝜑 → ∀𝑥𝜓)
9 vtocld.1 . 2 (𝜑𝐴𝑉)
10 vtoclgft 3552 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜒) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) ∧ ∀𝑥𝜓) ∧ 𝐴𝑉) → 𝜒)
111, 2, 6, 8, 9, 10syl221anc 1376 1 (𝜑𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1529   = wceq 1531  wnf 1778  wcel 2108  wnfc 2959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-ex 1775  df-nf 1779  df-cleq 2812  df-clel 2891  df-nfc 2961
This theorem is referenced by:  vtocld  3555  iota2df  6335  riotasv2d  36085
  Copyright terms: Public domain W3C validator