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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 3883 | . . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | 1 | abeq2i 2948 | . . 3 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵) |
3 | csbgfi.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | csbgfi.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcriv 2967 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
6 | 3, 5 | sbcgfi 3847 | . . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵) |
7 | 2, 6 | bitri 277 | . 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ 𝐵) |
8 | 7 | eqriv 2818 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 Ⅎwnfc 2961 Vcvv 3494 [wsbc 3771 ⦋csb 3882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-sbc 3772 df-csb 3883 |
This theorem is referenced by: fmptdF 30395 sbccom2f 35398 |
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