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Mirrors > Home > MPE Home > Th. List > 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 4333 | . . 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 2756 | . 2 ⊢ (⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐶 ↔ 𝐷 = 𝐸) |
7 | 3, 6 | bitri 278 | 1 ⊢ ([𝐴 / 𝑥]𝐵 = 𝐶 ↔ 𝐷 = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2111 Vcvv 3415 [wsbc 3703 ⦋csb 3820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-sbc 3704 df-csb 3821 |
This theorem is referenced by: sbccom2lem 36032 |
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