![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ensymd | GIF version |
Description: Symmetry of equinumerosity. Deduction form of ensym 6774. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ensymd.1 | ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Ref | Expression |
---|---|
ensymd | ⊢ (𝜑 → 𝐵 ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensymd.1 | . 2 ⊢ (𝜑 → 𝐴 ≈ 𝐵) | |
2 | ensym 6774 | . 2 ⊢ (𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → 𝐵 ≈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 class class class wbr 4000 ≈ cen 6731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-er 6528 df-en 6734 |
This theorem is referenced by: f1imaeng 6785 f1imaen2g 6786 en2sn 6806 xpdom3m 6827 phplem4 6848 phplem4dom 6855 php5dom 6856 phpm 6858 phplem4on 6860 dif1en 6872 dif1enen 6873 fisbth 6876 fin0 6878 fin0or 6879 fientri3 6907 unsnfidcex 6912 unsnfidcel 6913 fiintim 6921 fisseneq 6924 f1ofi 6935 endjusym 7088 eninl 7089 eninr 7090 pm54.43 7182 djuen 7203 dju1en 7205 djuassen 7209 xpdjuen 7210 uzenom 10398 hashennnuni 10730 hashennn 10731 hashcl 10732 hashfz1 10734 hashen 10735 fihashfn 10751 fihashdom 10754 hashunlem 10755 zfz1iso 10792 summodclem2 11361 zsumdc 11363 prodmodclem2 11556 zproddc 11558 ennnfonelemen 12392 exmidunben 12397 ctinfom 12399 ctinf 12401 pwf1oexmid 14371 sbthom 14397 |
Copyright terms: Public domain | W3C validator |