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

Theorem vtocldf 3254
 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 450 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜓𝜒)))
63, 5alrimi 2081 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)))
7 vtocld.3 . . 3 (𝜑𝜓)
83, 7alrimi 2081 . 2 (𝜑 → ∀𝑥𝜓)
9 vtocld.1 . 2 (𝜑𝐴𝑉)
10 vtoclgft 3252 . 2 (((𝑥𝐴 ∧ Ⅎ𝑥𝜒) ∧ (∀𝑥(𝑥 = 𝐴 → (𝜓𝜒)) ∧ ∀𝑥𝜓) ∧ 𝐴𝑉) → 𝜒)
111, 2, 6, 8, 9, 10syl221anc 1336 1 (𝜑𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1480   = wceq 1482  Ⅎwnf 1707   ∈ wcel 1989  Ⅎwnfc 2750 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-v 3200 This theorem is referenced by:  vtocld  3255  iota2df  5873  riotasv2d  34069
 Copyright terms: Public domain W3C validator