Theorem eqerlem 6523

Description: Lemma for eqer 6524. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Hypotheses

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

Assertion

Ref	Expression
eqerlem	⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)

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

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

Proof of Theorem eqerlem

Step	Hyp	Ref	Expression
1		eqer.2	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
2	1	brabsb 4233	. 2 ⊢ (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵)
3		vex 2724	. . 3 ⊢ 𝑧 ∈ V
4		nfcsb1v 3073	. . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴
5		nfcsb1v 3073	. . . . 5 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐴
6	4, 5	nfeq 2314	. . . 4 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴
7		vex 2724	. . . . . 6 ⊢ 𝑤 ∈ V
8		nfv 1515	. . . . . . 7 ⊢ Ⅎ𝑦 𝐴 = ⦋𝑤 / 𝑥⦌𝐴
9		vex 2724	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
10		nfcv 2306	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐵
11		eqer.1	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵)
12	9, 10, 11	csbief 3084	. . . . . . . . 9 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = 𝐵
13		csbeq1 3043	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
14	12, 13	eqtr3id 2211	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐴)
15	14	eqeq2d 2176	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴))
16	8, 15	sbciegf 2977	. . . . . 6 ⊢ (𝑤 ∈ V → ([𝑤 / 𝑦]𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴))
17	7, 16	ax-mp 5	. . . . 5 ⊢ ([𝑤 / 𝑦]𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
18		csbeq1a 3049	. . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴)
19	18	eqeq1d 2173	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴))
20	17, 19	syl5bb 191	. . . 4 ⊢ (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝐴 = 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴))
21	6, 20	sbciegf 2977	. . 3 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴))
22	3, 21	ax-mp 5	. 2 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)
23	2, 22	bitri 183	1 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1342   ∈ wcel 2135  Vcvv 2721  [wsbc 2946  ⦋csb 3040   class class class wbr 3976  {copab 4036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-v 2723  df-sbc 2947  df-csb 3041  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038

This theorem is referenced by:  eqer  6524