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Mirrors > Home > ILE Home > Th. List > sbceq2a | GIF version |
Description: Equality theorem for class substitution. Class version of sbequ12r 1746. (Contributed by NM, 4-Jan-2017.) |
Ref | Expression |
---|---|
sbceq2a | ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceq1a 2922 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | 1 | eqcoms 2143 | . 2 ⊢ (𝐴 = 𝑥 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
3 | 2 | bicomd 140 | 1 ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 [wsbc 2913 |
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 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-sbc 2914 |
This theorem is referenced by: (None) |
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