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

Theorem opelopabaf 4914
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4912 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypotheses
Ref Expression
opelopabaf.x 𝑥𝜓
opelopabaf.y 𝑦𝜓
opelopabaf.1 𝐴 ∈ V
opelopabaf.2 𝐵 ∈ V
opelopabaf.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
opelopabaf (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem opelopabaf
StepHypRef Expression
1 opelopabsb 4900 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
2 opelopabaf.1 . . 3 𝐴 ∈ V
3 opelopabaf.2 . . 3 𝐵 ∈ V
4 opelopabaf.x . . . 4 𝑥𝜓
5 opelopabaf.y . . . 4 𝑦𝜓
6 nfv 1829 . . . 4 𝑥 𝐵 ∈ V
7 opelopabaf.3 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
84, 5, 6, 7sbc2iegf 3470 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
92, 3, 8mp2an 703 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
101, 9bitri 262 1 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wnf 1698  wcel 1976  Vcvv 3172  [wsbc 3401  cop 4130  {copab 4636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator