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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 14635 | . 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 1537 〈“cs1 14630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-s1 14631 |
This theorem is referenced by: s1prc 14639 ccat1st1st 14663 swrds1 14701 swrdlsw 14702 reuccatpfxs1lem 14781 s2eqd 14899 s3eqd 14900 s4eqd 14901 s5eqd 14902 s6eqd 14903 s7eqd 14904 s8eqd 14905 frmdgsum 18888 psgnunilem5 19527 efgredlemc 19778 vrgpval 19800 vrgpinv 19802 frgpup2 19809 frgpup3lem 19810 pfx1s2 32908 pfxlsw2ccat 32920 ccatws1f1olast 32922 wrdpmtrlast 33096 1arithidomlem2 33544 iwrdsplit 34369 sseqval 34370 sseqf 34374 sseqp1 34377 signsvtn0 34564 signstfveq0 34571 mrsubcv 35495 |
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