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Theorem opelopabaf 5149
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 5147 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 5135 . 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 1992 . . . 4 𝑥 𝐵 ∈ V
7 opelopabaf.3 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
84, 5, 6, 7sbc2iegf 3645 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
92, 3, 8mp2an 710 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
101, 9bitri 264 1 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wnf 1857  wcel 2139  Vcvv 3340  [wsbc 3576  cop 4327  {copab 4864
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-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865
This theorem is referenced by: (None)
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