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 2254. (Contributed by NM, 4-Jan-2017.) |
Ref | Expression |
---|---|
sbceq2a | ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceq1a 3783 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
2 | 1 | eqcoms 2829 | . 2 ⊢ (𝐴 = 𝑥 → (𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
3 | 2 | bicomd 225 | 1 ⊢ (𝐴 = 𝑥 → ([𝐴 / 𝑥]𝜑 ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-sbc 3773 |
This theorem is referenced by: ralrnmptw 6860 tfindes 7577 rabssnn0fi 13355 indexa 35023 fdc 35035 fdc1 35036 alrimii 35412 tratrbVD 41215 |
Copyright terms: Public domain | W3C validator |