Mathbox for Giovanni Mascellani |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 3500 | . . . 4 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐵} | |
2 | 1 | abeq2i 2722 | . . 3 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ [𝐴 / 𝑥]𝑦 ∈ 𝐵) |
3 | csbgfi.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | csbgfi.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcri 2745 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
6 | 3, 5 | sbcgfi 33103 | . . 3 ⊢ ([𝐴 / 𝑥]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵) |
7 | 2, 6 | bitri 263 | . 2 ⊢ (𝑦 ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ 𝑦 ∈ 𝐵) |
8 | 7 | eqriv 2607 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 Vcvv 3173 [wsbc 3402 ⦋csb 3499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sbc 3403 df-csb 3500 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |