Theorem opelopabsbALT 4347

Description: The law of concretion in terms of substitutions. Less general than opelopabsb 4348, 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 1710	. . 3 ⊢ (∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑦∃𝑥(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
2		vex 2802	. . . . . . 7 ⊢ 𝑧 ∈ V
3		vex 2802	. . . . . . 7 ⊢ 𝑤 ∈ V
4	2, 3	opth 4323	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
5		equcom 1752	. . . . . . 7 ⊢ (𝑧 = 𝑥 ↔ 𝑥 = 𝑧)
6		equcom 1752	. . . . . . 7 ⊢ (𝑤 = 𝑦 ↔ 𝑦 = 𝑤)
7	5, 6	anbi12ci 461	. . . . . 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 458	. . . 4 ⊢ ((⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ 𝜑))
10	9	2exbii 1652	. . 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 4346	. 2 ⊢ (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦(⟨𝑧, 𝑤⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
13		2sb5 2034	. 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 1395  ∃wex 1538  [wsb 1808   ∈ wcel 2200  ⟨cop 3669  {copab 4144

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146

This theorem is referenced by:  inopab  4854  cnvopab  5130