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Mirrors > Home > ILE Home > Th. List > eqerlem | GIF version |
Description: Lemma for eqer 6391. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) |
Ref | Expression |
---|---|
eqer.1 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
eqer.2 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝐴 = 𝐵} |
Ref | Expression |
---|---|
eqerlem | ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqer.2 | . . 3 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝐴 = 𝐵} | |
2 | 1 | brabsb 4121 | . 2 ⊢ (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵) |
3 | vex 2644 | . . 3 ⊢ 𝑧 ∈ V | |
4 | nfcsb1v 2985 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴 | |
5 | nfcsb1v 2985 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐴 | |
6 | 4, 5 | nfeq 2248 | . . . 4 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 |
7 | vex 2644 | . . . . . 6 ⊢ 𝑤 ∈ V | |
8 | nfv 1476 | . . . . . . 7 ⊢ Ⅎ𝑦 𝐴 = ⦋𝑤 / 𝑥⦌𝐴 | |
9 | vex 2644 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
10 | nfcv 2240 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝐵 | |
11 | eqer.1 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
12 | 9, 10, 11 | csbief 2994 | . . . . . . . . 9 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = 𝐵 |
13 | csbeq1 2958 | . . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) | |
14 | 12, 13 | syl5eqr 2146 | . . . . . . . 8 ⊢ (𝑦 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐴) |
15 | 14 | eqeq2d 2111 | . . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
16 | 8, 15 | sbciegf 2892 | . . . . . 6 ⊢ (𝑤 ∈ V → ([𝑤 / 𝑦]𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
17 | 7, 16 | ax-mp 7 | . . . . 5 ⊢ ([𝑤 / 𝑦]𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
18 | csbeq1a 2963 | . . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | |
19 | 18 | eqeq1d 2108 | . . . . 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 2892 | . . 3 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
22 | 3, 21 | ax-mp 7 | . 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) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1299 ∈ wcel 1448 Vcvv 2641 [wsbc 2862 ⦋csb 2955 class class class wbr 3875 {copab 3928 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-rex 2381 df-v 2643 df-sbc 2863 df-csb 2956 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-br 3876 df-opab 3930 |
This theorem is referenced by: eqer 6391 |
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