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Mirrors > Home > ILE Home > Th. List > sbceq1dd | GIF version |
Description: Equality theorem for class substitution. (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by NM, 30-Jun-2018.) |
Ref | Expression |
---|---|
sbceq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
sbceq1dd.2 | ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) |
Ref | Expression |
---|---|
sbceq1dd | ⊢ (𝜑 → [𝐵 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceq1dd.2 | . 2 ⊢ (𝜑 → [𝐴 / 𝑥]𝜓) | |
2 | sbceq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 2 | sbceq1d 2769 | . 2 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓)) |
4 | 1, 3 | mpbid 135 | 1 ⊢ (𝜑 → [𝐵 / 𝑥]𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 [wsbc 2764 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-4 1400 ax-17 1419 ax-ial 1427 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-clel 2036 df-sbc 2765 |
This theorem is referenced by: (None) |
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