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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 13942 | . 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 1528 〈“cs1 13937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-s1 13938 |
This theorem is referenced by: s1prc 13946 ccat1st1st 13972 swrds1 14016 swrdlsw 14017 reuccatpfxs1lem 14096 s2eqd 14213 s3eqd 14214 s4eqd 14215 s5eqd 14216 s6eqd 14217 s7eqd 14218 s8eqd 14219 frmdgsum 18015 psgnunilem5 18551 efgredlemc 18800 vrgpval 18822 vrgpinv 18824 frgpup2 18831 frgpup3lem 18832 pfx1s2 30542 pfxlsw2ccat 30553 iwrdsplit 31544 sseqval 31545 sseqf 31549 sseqp1 31552 signsvtn0 31739 signstfveq0 31746 mrsubcv 32654 |
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