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Mirrors > Home > ILE Home > Th. List > ensymd | GIF version |
Description: Symmetry of equinumerosity. Deduction form of ensym 6802. (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 6802 | . 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 4018 ≈ cen 6759 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-er 6554 df-en 6762 |
This theorem is referenced by: f1imaeng 6813 f1imaen2g 6814 en2sn 6834 xpdom3m 6855 phplem4 6878 phplem4dom 6885 php5dom 6886 phpm 6888 phplem4on 6890 dif1en 6902 dif1enen 6903 fisbth 6906 fin0 6908 fin0or 6909 fientri3 6938 unsnfidcex 6943 unsnfidcel 6944 fiintim 6952 fisseneq 6955 f1ofi 6967 endjusym 7120 eninl 7121 eninr 7122 pm54.43 7214 djuen 7235 dju1en 7237 djuassen 7241 xpdjuen 7242 uzenom 10451 hashennnuni 10786 hashennn 10787 hashcl 10788 hashfz1 10790 hashen 10791 fihashfn 10807 fihashdom 10810 hashunlem 10811 zfz1iso 10848 summodclem2 11417 zsumdc 11419 prodmodclem2 11612 zproddc 11614 4sqlem11 12428 ennnfonelemen 12467 exmidunben 12472 ctinfom 12474 ctinf 12476 pwf1oexmid 15188 sbthom 15213 |
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