MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtocl2 Structured version   Visualization version   GIF version

Theorem vtocl2 3401
Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
vtocl2.1 𝐴 ∈ V
vtocl2.2 𝐵 ∈ V
vtocl2.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
vtocl2.4 𝜑
Assertion
Ref Expression
vtocl2 𝜓
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem vtocl2
StepHypRef Expression
1 vtocl2.1 . . . . . 6 𝐴 ∈ V
21isseti 3349 . . . . 5 𝑥 𝑥 = 𝐴
3 vtocl2.2 . . . . . 6 𝐵 ∈ V
43isseti 3349 . . . . 5 𝑦 𝑦 = 𝐵
5 eeanv 2327 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6 vtocl2.3 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
76biimpd 219 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
872eximi 1912 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦(𝜑𝜓))
95, 8sylbir 225 . . . . 5 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥𝑦(𝜑𝜓))
102, 4, 9mp2an 710 . . . 4 𝑥𝑦(𝜑𝜓)
11 19.36v 2069 . . . . 5 (∃𝑦(𝜑𝜓) ↔ (∀𝑦𝜑𝜓))
1211exbii 1923 . . . 4 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(∀𝑦𝜑𝜓))
1310, 12mpbi 220 . . 3 𝑥(∀𝑦𝜑𝜓)
141319.36iv 2023 . 2 (∀𝑥𝑦𝜑𝜓)
15 vtocl2.4 . . 3 𝜑
1615ax-gen 1871 . 2 𝑦𝜑
1714, 16mpg 1873 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-v 3342
This theorem is referenced by:  caovord  7010  sornom  9291  wloglei  10752  ipodrsima  17366  mpfind  19738  mclsppslem  31787  monotoddzzfi  38009
  Copyright terms: Public domain W3C validator