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Theorem opelopabaf 5225
 Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 5223 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 5211 . 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 2015 . . . 4 𝑥 𝐵 ∈ V
7 opelopabaf.3 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
84, 5, 6, 7sbc2iegf 3729 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓))
92, 3, 8mp2an 685 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑𝜓)
101, 9bitri 267 1 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658  Ⅎwnf 1884   ∈ wcel 2166  Vcvv 3414  [wsbc 3662  ⟨cop 4403  {copab 4935 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-opab 4936 This theorem is referenced by: (None)
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