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Mirrors > Home > MPE Home > Th. List > eqerlem | Structured version Visualization version GIF version |
Description: Lemma for eqer 8684. (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 5489 | . 2 ⊢ (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵) |
3 | nfcsb1v 3881 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴 | |
4 | nfcsb1v 3881 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑤 / 𝑥⦌𝐴 | |
5 | 3, 4 | nfeq 2921 | . . . 4 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴 |
6 | nfv 1918 | . . . . . . 7 ⊢ Ⅎ𝑦 𝐴 = ⦋𝑤 / 𝑥⦌𝐴 | |
7 | vex 3450 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
8 | eqer.1 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
9 | 7, 8 | csbie 3892 | . . . . . . . . 9 ⊢ ⦋𝑦 / 𝑥⦌𝐴 = 𝐵 |
10 | csbeq1 3859 | . . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) | |
11 | 9, 10 | eqtr3id 2791 | . . . . . . . 8 ⊢ (𝑦 = 𝑤 → 𝐵 = ⦋𝑤 / 𝑥⦌𝐴) |
12 | 11 | eqeq2d 2748 | . . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
13 | 6, 12 | sbciegf 3779 | . . . . . 6 ⊢ (𝑤 ∈ V → ([𝑤 / 𝑦]𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
14 | 13 | elv 3452 | . . . . 5 ⊢ ([𝑤 / 𝑦]𝐴 = 𝐵 ↔ 𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
15 | csbeq1a 3870 | . . . . . 6 ⊢ (𝑥 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑥⦌𝐴) | |
16 | 15 | eqeq1d 2739 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝐴 = ⦋𝑤 / 𝑥⦌𝐴 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
17 | 14, 16 | bitrid 283 | . . . 4 ⊢ (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝐴 = 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
18 | 5, 17 | sbciegf 3779 | . . 3 ⊢ (𝑧 ∈ V → ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴)) |
19 | 18 | elv 3452 | . 2 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
20 | 2, 19 | bitri 275 | 1 ⊢ (𝑧𝑅𝑤 ↔ ⦋𝑧 / 𝑥⦌𝐴 = ⦋𝑤 / 𝑥⦌𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 Vcvv 3446 [wsbc 3740 ⦋csb 3856 class class class wbr 5106 {copab 5168 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 |
This theorem is referenced by: eqer 8684 |
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