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Mirrors > Home > MPE Home > Th. List > sbceq2a | Structured version Visualization version GIF version |
Description: Equality theorem for class substitution. Class version of sbequ12r 2239. (Contributed by NM, 4-Jan-2017.) |
Ref | Expression |
---|---|
sbceq2a | ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceq1a 3784 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | 1 | eqcoms 2733 | . 2 ⊢ (𝐴 = 𝑥 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
3 | 2 | bicomd 222 | 1 ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 [wsbc 3773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-sbc 3774 |
This theorem is referenced by: ralrnmptw 7103 tfindes 7868 rabssnn0fi 13992 indexa 37339 fdc 37351 fdc1 37352 alrimii 37725 tratrbVD 44444 |
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