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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 2740 | . . 3 ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒}) |
5 | 1, 4 | eleq12d 2695 | . 2 ⊢ (𝜑 → (𝐴 ∈ {𝑥 ∣ 𝜓} ↔ 𝐵 ∈ {𝑥 ∣ 𝜒})) |
6 | df-sbc 3436 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 ↔ 𝐴 ∈ {𝑥 ∣ 𝜓}) | |
7 | df-sbc 3436 | . 2 ⊢ ([𝐵 / 𝑥]𝜒 ↔ 𝐵 ∈ {𝑥 ∣ 𝜒}) | |
8 | 5, 6, 7 | 3bitr4g 303 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 {cab 2608 [wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-sbc 3436 |
This theorem is referenced by: (None) |
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