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Mirrors > Home > MPE Home > Th. List > eqerlem | Structured version Visualization version GIF version |
Description: Lemma for eqer 8766. (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 5537 | . 2 ⊢ (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵) |
3 | nfcsb1v 3919 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴 | |
4 | nfcsb1v 3919 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐴 | |
5 | 3, 4 | nfeq 2913 | . . . 4 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 |
6 | nfv 1909 | . . . . . . 7 ⊢ Ⅎ𝑦 𝐴 = ⦋𝑤 / 𝑥⦌𝐴 | |
7 | vex 3477 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
8 | eqer.1 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
9 | 7, 8 | csbie 3930 | . . . . . . . . 9 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = 𝐵 |
10 | csbeq1 3897 | . . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) | |
11 | 9, 10 | eqtr3id 2782 | . . . . . . . 8 ⊢ (𝑦 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐴) |
12 | 11 | eqeq2d 2739 | . . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
13 | 6, 12 | sbciegf 3818 | . . . . . 6 ⊢ (𝑤 ∈ V → ([𝑤 / 𝑦]𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
14 | 13 | elv 3479 | . . . . 5 ⊢ ([𝑤 / 𝑦]𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
15 | csbeq1a 3908 | . . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | |
16 | 15 | eqeq1d 2730 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
17 | 14, 16 | bitrid 282 | . . . 4 ⊢ (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝐴 = 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
18 | 5, 17 | sbciegf 3818 | . . 3 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
19 | 18 | elv 3479 | . 2 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
20 | 2, 19 | bitri 274 | 1 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 Vcvv 3473 [wsbc 3778 ⦋csb 3894 class class class wbr 5152 {copab 5214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 |
This theorem is referenced by: eqer 8766 |
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