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Mirrors > Home > MPE Home > Th. List > sbceq1g | Structured version Visualization version GIF version |
Description: Move proper substitution to first argument of an equality. (Contributed by NM, 30-Nov-2005.) |
Ref | Expression |
---|---|
sbceq1g | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceqg 4358 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) | |
2 | csbconstg 3899 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐶) | |
3 | 2 | eqeq2d 2829 | . 2 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
4 | 1, 3 | bitrd 280 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 [wsbc 3769 ⦋csb 3880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-sbc 3770 df-csb 3881 |
This theorem is referenced by: telgsums 19042 suppss2f 30312 f1od2 30383 finxpreclem4 34557 frege120 40207 |
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