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Mirrors > Home > ILE Home > Th. List > sbceq1d | 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) |
Ref | Expression |
---|---|
sbceq1d | ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | dfsbcq 2906 | . 2 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ [𝐵 / 𝑥]𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 [wsbc 2904 |
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 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-17 1506 ax-ial 1514 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-cleq 2130 df-clel 2133 df-sbc 2905 |
This theorem is referenced by: sbceq1dd 2910 rexrnmpt 5556 findcard2 6776 findcard2s 6777 ac6sfi 6785 nn1suc 8732 uzind4s 9378 uzind4s2 9379 fzrevral 9878 fzshftral 9881 cjth 10611 prmind2 11790 bj-bdfindes 13136 bj-findes 13168 |
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