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Mirrors > Home > MPE Home > Th. List > necon3bbid | Structured version Visualization version GIF version |
Description: Deduction from equality to inequality. (Contributed by NM, 2-Jun-2007.) |
Ref | Expression |
---|---|
necon3bbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝐴 = 𝐵)) |
Ref | Expression |
---|---|
necon3bbid | ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon3bbid.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝐴 = 𝐵)) | |
2 | 1 | bicomd 212 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓)) |
3 | 2 | necon3abid 2818 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓)) |
4 | 3 | bicomd 212 | 1 ⊢ (𝜑 → (¬ 𝜓 ↔ 𝐴 ≠ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 = wceq 1475 ≠ wne 2780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 df-ne 2782 |
This theorem is referenced by: necon1abid 2820 necon3bid 2826 eldifsn 4260 php 8029 xmullem2 11967 seqcoll2 13106 cnpart 13828 rlimrecl 14159 ncoprmgcdne1b 15201 prmrp 15262 4sqlem17 15503 mrieqvd 16121 mrieqv2d 16122 pltval 16783 latnlemlt 16907 latnle 16908 odnncl 17787 gexnnod 17826 sylow1lem1 17836 slwpss 17850 lssnle 17910 lspsnne1 18938 nzrunit 19088 psrridm 19225 cnsubrg 19625 cmpfi 21021 hausdiag 21258 txhaus 21260 isusp 21875 recld2 22425 metdseq0 22465 i1f1lem 23262 aaliou2b 23900 dvloglem 24194 logf1o2 24196 lgsne0 24860 lgsqr 24876 2sqlem7 24949 ostth3 25127 tglngne 25245 tgelrnln 25325 norm1exi 27491 atnemeq0 28620 opeldifid 28794 qtophaus 29231 ordtconlem1 29298 elzrhunit 29351 sgnneg 29929 subfacp1lem6 30421 maxidln1 33013 smprngopr 33021 lsatnem0 33350 atncmp 33617 atncvrN 33620 cdlema2N 34096 lhpmatb 34335 lhpat3 34350 cdleme3 34542 cdleme7 34554 cdlemg27b 35002 dvh2dimatN 35747 dvh2dim 35752 dochexmidlem1 35767 dochfln0 35784 eucrct2eupth 41413 |
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