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Mirrors > Home > ILE Home > Th. List > sbsbc | GIF version |
Description: Show that df-sb 1717 and df-sbc 2877 are equivalent when the class term 𝐴 in df-sbc 2877 is a setvar variable. This theorem lets us reuse theorems based on df-sb 1717 for proofs involving df-sbc 2877. (Contributed by NM, 31-Dec-2016.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
sbsbc | ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2113 | . 2 ⊢ 𝑦 = 𝑦 | |
2 | dfsbcq2 2879 | . 2 ⊢ (𝑦 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 [wsb 1716 [wsbc 2876 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1404 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-4 1468 ax-17 1487 ax-ial 1495 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-clab 2100 df-cleq 2106 df-clel 2109 df-sbc 2877 |
This theorem is referenced by: spsbc 2887 sbcid 2891 sbcco 2897 sbcco2 2898 sbcie2g 2908 eqsbc3 2914 sbcralt 2951 sbcrext 2952 sbnfc2 3024 csbabg 3025 cbvralcsf 3026 cbvrexcsf 3027 cbvreucsf 3028 cbvrabcsf 3029 isarep1 5165 finexdc 6747 ssfirab 6772 zsupcllemstep 11480 bezoutlemmain 11526 bdsbc 12739 |
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