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Mathbox for Giovanni Mascellani |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbceqi | Structured version Visualization version GIF version |
Description: Distribution of class substitution over equality, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.) |
Ref | Expression |
---|---|
sbceqi.1 | ⊢ 𝐴 ∈ V |
sbceqi.2 | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷 |
sbceqi.3 | ⊢ ⦋𝐴 / 𝑥⦌𝐶 = 𝐸 |
Ref | Expression |
---|---|
sbceqi | ⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐷 = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceqi.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | sbceqg 4128 | . . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶) |
4 | sbceqi.2 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐷 | |
5 | sbceqi.3 | . . 3 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = 𝐸 | |
6 | 4, 5 | eqeq12i 2785 | . 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐷 = 𝐸) |
7 | 3, 6 | bitri 264 | 1 ⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐷 = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1631 ∈ wcel 2145 Vcvv 3351 [wsbc 3587 ⦋csb 3682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-v 3353 df-sbc 3588 df-csb 3683 |
This theorem is referenced by: sbccom2lem 34257 |
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