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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 3854 | . . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | 1 | abeq2i 2874 | . . 3 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵) |
3 | csbgfi.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | csbgfi.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcri 2892 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
6 | 3, 5 | sbcgfi 3818 | . . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵) |
7 | 2, 6 | bitri 274 | . 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ 𝐵) |
8 | 7 | eqriv 2733 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 Ⅎwnfc 2885 Vcvv 3443 [wsbc 3737 ⦋csb 3853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-sbc 3738 df-csb 3854 |
This theorem is referenced by: fmptdF 31417 sbccom2f 36517 |
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