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Mirrors > Home > MPE Home > Th. List > Mathboxes > sbceqbidf | Structured version Visualization version GIF version |
Description: Equality theorem for class substitution. (Contributed by Thierry Arnoux, 4-Sep-2018.) |
Ref | Expression |
---|---|
sbceqbidf.1 | ⊢ Ⅎ𝑥𝜑 |
sbceqbidf.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
sbceqbidf.3 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
sbceqbidf | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceqbidf.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | sbceqbidf.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
3 | sbceqbidf.3 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
4 | 2, 3 | abbid 2887 | . . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
5 | 1, 4 | eleq12d 2907 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐵 ∈ {𝑥 ∣ 𝜒})) |
6 | df-sbc 3772 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}) | |
7 | df-sbc 3772 | . 2 ⊢ ([𝐵 / 𝑥]𝜒 ↔ 𝐵 ∈ {𝑥 ∣ 𝜒}) | |
8 | 5, 6, 7 | 3bitr4g 316 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 {cab 2799 [wsbc 3771 |
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-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-sbc 3772 |
This theorem is referenced by: (None) |
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