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Mirrors > Home > MPE Home > Th. List > nfeq | Structured version Visualization version GIF version |
Description: Hypothesis builder for equality. (Contributed by NM, 21-Jun-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) |
Ref | Expression |
---|---|
nfnfc.1 | ⊢ Ⅎ𝑥𝐴 |
nfeq.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfeq | ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfnfc.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
3 | nfeq.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐵) |
5 | 2, 4 | nfeqd 2913 | . 2 ⊢ (⊤ → Ⅎ𝑥 𝐴 = 𝐵) |
6 | 5 | mptru 1543 | 1 ⊢ Ⅎ𝑥 𝐴 = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ⊤wtru 1537 Ⅎwnf 1779 Ⅎwnfc 2887 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1539 df-ex 1776 df-nf 1780 df-cleq 2726 df-nfc 2889 |
This theorem is referenced by: nfeq1 2918 nfeq2 2920 nfne 3040 raleqf 3350 rexeqfOLD 3352 rmoeq1f 3420 rabeqf 3469 csbhypf 3936 sbceqg 4417 nffn 6667 nffo 6819 fvmptd3f 7030 mpteqb 7034 fvmptf 7036 eqfnfv2f 7054 dff13f 7275 ovmpos 7580 ov2gf 7581 ovmpodxf 7582 ovmpodf 7588 eqerlem 8778 seqof2 14097 sumeq2ii 15725 sumss 15756 fsumadd 15772 fsummulc2 15816 fsumrelem 15839 prodeq1f 15938 prodeq2ii 15943 fprodmul 15992 fproddiv 15993 txcnp 23643 ptcnplem 23644 cnmpt11 23686 cnmpt21 23694 cnmptcom 23701 mbfeqalem1 25689 mbflim 25716 itgeq1f 25820 itgeq1fOLD 25821 itgeqa 25863 dvmptfsum 26027 ulmss 26454 leibpi 26999 o1cxp 27032 lgseisenlem2 27434 nosupbnd1 27773 2ndresdju 32665 aciunf1lem 32678 sigapildsys 34142 bnj1316 34812 bnj1446 35037 bnj1447 35038 bnj1448 35039 bnj1519 35057 bnj1520 35058 bnj1529 35062 subtr 36296 subtr2 36297 bj-sbeqALT 36882 poimirlem25 37631 iuneq2f 38142 mpobi123f 38148 mptbi12f 38152 dvdsrabdioph 42797 fphpd 42803 mnringmulrcld 44223 fvelrnbf 44955 refsum2cnlem1 44974 elrnmpt1sf 45131 choicefi 45142 axccdom 45164 uzublem 45379 fsumf1of 45529 fmuldfeq 45538 mccl 45553 climmulf 45559 climexp 45560 climsuse 45563 climrecf 45564 climaddf 45570 mullimc 45571 neglimc 45602 addlimc 45603 0ellimcdiv 45604 climeldmeqmpt 45623 climfveqmpt 45626 climfveqf 45635 climfveqmpt3 45637 climeldmeqf 45638 climeqf 45643 climeldmeqmpt3 45644 limsupubuzlem 45667 limsupequz 45678 dvnmptdivc 45893 dvmptfprod 45900 stoweidlem18 45973 stoweidlem31 45986 stoweidlem55 46010 stoweidlem59 46014 sge0iunmpt 46373 sge0reuz 46402 iundjiun 46415 hoicvrrex 46511 ovnhoilem1 46556 ovnlecvr2 46565 opnvonmbllem1 46587 vonioo 46637 vonicc 46640 sssmf 46693 smflim 46732 smfpimcclem 46762 smfpimcc 46763 cfsetsnfsetf 47007 ovmpordxf 48183 |
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