Theorem eqerOLD 7665

Description: Obsolete proof of eqer 7664 as of 1-May-2021. Equivalence relation involving equality of dependent classes 𝐴(𝑥) and 𝐵(𝑦). (Contributed by NM, 17-Mar-2008.) (Revised by Mario Carneiro, 12-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
eqer.1	⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
eqer.2	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}

Assertion

Ref	Expression
eqerOLD	⊢ 𝑅 Er V

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem eqerOLD

Dummy variables 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqer.2	. . . . 5 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
2	1	relopabi 5167	. . . 4 ⊢ Rel 𝑅
3	2	a1i 11	. . 3 ⊢ (⊤ → Rel 𝑅)
4		id 22	. . . . . 6 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
5	4	eqcomd 2616	. . . . 5 ⊢ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 → ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
6		eqer.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
7	6, 1	eqerlem 7663	. . . . 5 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
8	6, 1	eqerlem 7663	. . . . 5 ⊢ (𝑤𝑅𝑧 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
9	5, 7, 8	3imtr4i 280	. . . 4 ⊢ (𝑧𝑅𝑤 → 𝑤𝑅𝑧)
10	9	adantl 481	. . 3 ⊢ ((⊤ ∧ 𝑧𝑅𝑤) → 𝑤𝑅𝑧)
11		eqtr 2629	. . . . 5 ⊢ ((⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴) → ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
12	6, 1	eqerlem 7663	. . . . . 6 ⊢ (𝑤𝑅𝑣 ↔ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
13	7, 12	anbi12i 729	. . . . 5 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) ↔ (⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ∧ ⦋𝑤 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴))
14	6, 1	eqerlem 7663	. . . . 5 ⊢ (𝑧𝑅𝑣 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑣 / 𝑥⦌𝐴)
15	11, 13, 14	3imtr4i 280	. . . 4 ⊢ ((𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣) → 𝑧𝑅𝑣)
16	15	adantl 481	. . 3 ⊢ ((⊤ ∧ (𝑧𝑅𝑤 ∧ 𝑤𝑅𝑣)) → 𝑧𝑅𝑣)
17		vex 3176	. . . . 5 ⊢ 𝑧 ∈ V
18		eqid 2610	. . . . . 6 ⊢ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴
19	6, 1	eqerlem 7663	. . . . . 6 ⊢ (𝑧𝑅𝑧 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
20	18, 19	mpbir 220	. . . . 5 ⊢ 𝑧𝑅𝑧
21	17, 20	2th 253	. . . 4 ⊢ (𝑧 ∈ V ↔ 𝑧𝑅𝑧)
22	21	a1i 11	. . 3 ⊢ (⊤ → (𝑧 ∈ V ↔ 𝑧𝑅𝑧))
23	3, 10, 16, 22	iserd 7655	. 2 ⊢ (⊤ → 𝑅 Er V)
24	23	trud 1484	1 ⊢ 𝑅 Er V

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499   class class class wbr 4583  {copab 4642  Rel wrel 5043   Er wer 7626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-er 7629

This theorem is referenced by: (None)