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| Mirrors > Home > MPE Home > Th. List > csbgfi | Structured version Visualization version GIF version | ||
| Description: Substitution for a variable not free in a class does not affect it, in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.) |
| Ref | Expression |
|---|---|
| csbgfi.1 | ⊢ 𝐴 ∈ V |
| csbgfi.2 | ⊢ Ⅎ𝑥𝐵 |
| Ref | Expression |
|---|---|
| csbgfi | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐵 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-csb 3900 | . . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
| 2 | 1 | eqabri 2885 | . . 3 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵) |
| 3 | csbgfi.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 4 | csbgfi.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
| 5 | 4 | nfcri 2897 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
| 6 | 3, 5 | sbcgfi 3864 | . . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵) |
| 7 | 2, 6 | bitri 275 | . 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ 𝐵) |
| 8 | 7 | eqriv 2734 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 Vcvv 3480 [wsbc 3788 ⦋csb 3899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-sbc 3789 df-csb 3900 |
| This theorem is referenced by: fmptdF 32666 sbccom2f 38133 evl1gprodd 42118 |
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