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Theorem opelopabsbALT 5413
Description: The law of concretion in terms of substitutions. Less general than opelopabsb 5414, 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 2161 . . 3 (∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2 vex 3503 . . . . . . 7 𝑧 ∈ V
3 vex 3503 . . . . . . 7 𝑤 ∈ V
42, 3opth 5365 . . . . . 6 (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = 𝑥𝑤 = 𝑦))
5 equcom 2018 . . . . . . 7 (𝑧 = 𝑥𝑥 = 𝑧)
6 equcom 2018 . . . . . . 7 (𝑤 = 𝑦𝑦 = 𝑤)
75, 6anbi12ci 627 . . . . . 6 ((𝑧 = 𝑥𝑤 = 𝑦) ↔ (𝑦 = 𝑤𝑥 = 𝑧))
84, 7bitri 276 . . . . 5 (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = 𝑤𝑥 = 𝑧))
98anbi1i 623 . . . 4 ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
1092exbii 1842 . . 3 (∃𝑦𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
111, 10bitri 276 . 2 (∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
12 elopab 5411 . 2 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
13 2sb5 2276 . 2 ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ ∃𝑦𝑥((𝑦 = 𝑤𝑥 = 𝑧) ∧ 𝜑))
1411, 12, 133bitr4i 304 1 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wex 1773  [wsb 2062  wcel 2107  cop 4570  {copab 5125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-opab 5126
This theorem is referenced by:  inopab  5700  cnvopab  5995  brabsb2  35865
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