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Mirrors > Home > ILE Home > Th. List > ensymd | GIF version |
Description: Symmetry of equinumerosity. Deduction form of ensym 6605. (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 6605 | . 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 3875 ≈ cen 6562 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-er 6359 df-en 6565 |
This theorem is referenced by: f1imaeng 6616 f1imaen2g 6617 en2sn 6637 xpdom3m 6657 phplem4 6678 phplem4dom 6685 php5dom 6686 phpm 6688 phplem4on 6690 dif1en 6702 dif1enen 6703 fisbth 6706 fin0 6708 fin0or 6709 fientri3 6732 unsnfidcex 6737 unsnfidcel 6738 fiintim 6746 fisseneq 6749 f1ofi 6759 endjusym 6896 eninl 6897 eninr 6898 pm54.43 6957 djuen 6971 dju1en 6973 djuassen 6977 xpdjuen 6978 uzenom 10039 hashennnuni 10366 hashennn 10367 hashcl 10368 hashfz1 10370 hashen 10371 fihashfn 10387 fihashdom 10390 hashunlem 10391 zfz1iso 10425 summodclem2 10990 zsumdc 10992 ennnfonelemen 11726 exmidunben 11731 ctinfom 11733 ctinf 11735 pwf1oexmid 12780 sbthom 12805 |
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