Theorem opelopabsbALT 4396

Description: The law of concretion in terms of substitutions. Less general than opelopabsb 4397, 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

Step	Hyp	Ref	Expression
1		excom 1716	. . 3 ⊢ (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2		vex 2824	. . . . . . 7 ⊢ 𝑧 ∈ V
3		vex 2824	. . . . . . 7 ⊢ 𝑤 ∈ V
4	2, 3	opth 4372	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
5		equcom 1758	. . . . . . 7 ⊢ (𝑧 = 𝑥 ↔ 𝑥 = 𝑧)
6		equcom 1758	. . . . . . 7 ⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤)
7	5, 6	anbi12ci 465	. . . . . 6 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ↔ (𝑦 = 𝑤 ∧ 𝑥 = 𝑧))
8	4, 7	bitri 184	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = 𝑤 ∧ 𝑥 = 𝑧))
9	8	anbi1i 462	. . . 4 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑))
10	9	2exbii 1659	. . 3 ⊢ (∃𝑦∃𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑))
11	1, 10	bitri 184	. 2 ⊢ (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑))
12		elopab 4395	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
13		2sb5 2043	. 2 ⊢ ([𝑤 / 𝑦][𝑧 / 𝑥]𝜑 ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑))
14	11, 12, 13	3bitr4i 212	1 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545  [wsb 1815   ∈ wcel 2209  ⟨cop 3708  {copab 4186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-opab 4188

This theorem is referenced by:  inopab  4907  cnvopab  5184