![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > s1eqd | Structured version Visualization version GIF version |
Description: Equality theorem for a singleton word. (Contributed by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
s1eqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
s1eqd | ⊢ (𝜑 → 〈“𝐴”〉 = 〈“𝐵”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1eqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | s1eq 14603 | . 2 ⊢ (𝐴 = 𝐵 → 〈“𝐴”〉 = 〈“𝐵”〉) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 〈“𝐴”〉 = 〈“𝐵”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 〈“cs1 14598 |
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-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4325 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-s1 14599 |
This theorem is referenced by: s1prc 14607 ccat1st1st 14631 swrds1 14669 swrdlsw 14670 reuccatpfxs1lem 14749 s2eqd 14867 s3eqd 14868 s4eqd 14869 s5eqd 14870 s6eqd 14871 s7eqd 14872 s8eqd 14873 frmdgsum 18847 psgnunilem5 19487 efgredlemc 19738 vrgpval 19760 vrgpinv 19762 frgpup2 19769 frgpup3lem 19770 pfx1s2 32791 pfxlsw2ccat 32802 ccatws1f1olast 32804 wrdpmtrlast 32948 1arithidomlem2 33388 iwrdsplit 34177 sseqval 34178 sseqf 34182 sseqp1 34185 signsvtn0 34372 signstfveq0 34379 mrsubcv 35290 |
Copyright terms: Public domain | W3C validator |