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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 2951 | . 2 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓)) |
4 | 1, 3 | mpbid 146 | 1 ⊢ (𝜑 → [𝐵 / 𝑥]𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 [wsbc 2946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1434 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-4 1497 ax-17 1513 ax-ial 1521 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-cleq 2157 df-clel 2160 df-sbc 2947 |
This theorem is referenced by: prmind2 12031 |
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