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Mirrors > Home > ILE Home > Th. List > eqerlem | GIF version |
Description: Lemma for eqer 6138. (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 3998 | . 2 ⊢ (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵) |
3 | vex 2560 | . . 3 ⊢ 𝑧 ∈ V | |
4 | nfcsb1v 2882 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴 | |
5 | nfcsb1v 2882 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐴 | |
6 | 4, 5 | nfeq 2185 | . . . 4 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 |
7 | vex 2560 | . . . . . 6 ⊢ 𝑤 ∈ V | |
8 | nfv 1421 | . . . . . . 7 ⊢ Ⅎ𝑦 𝐴 = ⦋𝑤 / 𝑥⦌𝐴 | |
9 | vex 2560 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
10 | nfcv 2178 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝐵 | |
11 | eqer.1 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
12 | 9, 10, 11 | csbief 2891 | . . . . . . . . 9 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = 𝐵 |
13 | csbeq1 2855 | . . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) | |
14 | 12, 13 | syl5eqr 2086 | . . . . . . . 8 ⊢ (𝑦 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐴) |
15 | 14 | eqeq2d 2051 | . . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
16 | 8, 15 | sbciegf 2794 | . . . . . 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 2860 | . . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | |
19 | 18 | eqeq1d 2048 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
20 | 17, 19 | syl5bb 181 | . . . 4 ⊢ (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝐴 = 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
21 | 6, 20 | sbciegf 2794 | . . 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 173 | 1 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1243 ∈ wcel 1393 Vcvv 2557 [wsbc 2764 ⦋csb 2852 class class class wbr 3764 {copab 3817 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 |
This theorem is referenced by: eqer 6138 |
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