![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sbceq2a | Structured version Visualization version GIF version |
Description: Equality theorem for class substitution. Class version of sbequ12r 2267. (Contributed by NM, 4-Jan-2017.) |
Ref | Expression |
---|---|
sbceq2a | ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceq1a 3596 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | 1 | eqcoms 2778 | . 2 ⊢ (𝐴 = 𝑥 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
3 | 2 | bicomd 213 | 1 ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1630 [wsbc 3585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-12 2202 ax-ext 2750 |
This theorem depends on definitions: df-bi 197 df-an 383 df-ex 1852 df-sb 2049 df-clab 2757 df-cleq 2763 df-clel 2766 df-sbc 3586 |
This theorem is referenced by: tfindes 7208 rabssnn0fi 12992 indexa 33853 fdc 33866 fdc1 33867 alrimii 34249 tratrbVD 39613 |
Copyright terms: Public domain | W3C validator |