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Theorem opelopabsbALT 5178
Description: The law of concretion in terms of substitutions. Less general than opelopabsb 5179, but having a much shorter proof. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
opelopabsbALT (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝑤,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem opelopabsbALT
StepHypRef Expression
1 excom 2205 . . 3 (∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 vex 3386 . . . . . . 7 𝑧 ∈ V
3 vex 3386 . . . . . . 7 𝑤 ∈ V
42, 3opth 5133 . . . . . 6 (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = 𝑥𝑤 = 𝑦))
5 equcom 2117 . . . . . . 7 (𝑧 = 𝑥𝑥 = 𝑧)
6 equcom 2117 . . . . . . 7 (𝑤 = 𝑦𝑦 = 𝑤)
75, 6anbi12ci 622 . . . . . 6 ((𝑧 = 𝑥𝑤 = 𝑦) ↔ (𝑦 = 𝑤𝑥 = 𝑧))
84, 7bitri 267 . . . . 5 (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = 𝑤𝑥 = 𝑧))
98anbi1i 618 . . . 4 ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
1092exbii 1945 . . 3 (∃𝑦𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
111, 10bitri 267 . 2 (∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
12 elopab 5177 . 2 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
13 2sb5 2301 . 2 ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
1411, 12, 133bitr4i 295 1 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wex 1875  [wsb 2064  wcel 2157  cop 4372  {copab 4903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-opab 4904
This theorem is referenced by:  inopab  5454  cnvopab  5749  brabsb2  34874
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