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Mirrors > Home > ILE Home > Th. List > nfcxfr | GIF version |
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfceqi.1 | ⊢ 𝐴 = 𝐵 |
nfcxfr.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfcxfr | ⊢ Ⅎ𝑥𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcxfr.2 | . 2 ⊢ Ⅎ𝑥𝐵 | |
2 | nfceqi.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 2 | nfceqi 2254 | . 2 ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵) |
4 | 1, 3 | mpbir 145 | 1 ⊢ Ⅎ𝑥𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 Ⅎwnfc 2245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-17 1491 ax-ial 1499 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-cleq 2110 df-clel 2113 df-nfc 2247 |
This theorem is referenced by: nfrab1 2587 nfrabxy 2588 nfdif 3167 nfun 3202 nfin 3252 nfpw 3493 nfpr 3543 nfsn 3553 nfop 3691 nfuni 3712 nfint 3751 nfiunxy 3809 nfiinxy 3810 nfiunya 3811 nfiinya 3812 nfiu1 3813 nfii1 3814 nfopab 3966 nfopab1 3967 nfopab2 3968 nfmpt 3990 nfmpt1 3991 repizf2 4056 nfsuc 4300 nfxp 4536 nfco 4674 nfcnv 4688 nfdm 4753 nfrn 4754 nfres 4791 nfima 4859 nfiota1 5060 nffv 5399 fvmptss2 5464 fvmptssdm 5473 fvmptf 5481 ralrnmpt 5530 rexrnmpt 5531 f1ompt 5539 f1mpt 5640 fliftfun 5665 nfriota1 5705 riotaprop 5721 nfoprab1 5788 nfoprab2 5789 nfoprab3 5790 nfoprab 5791 nfmpo1 5806 nfmpo2 5807 nfmpo 5808 ovmpos 5862 ov2gf 5863 ovi3 5875 nfof 5955 nfofr 5956 nftpos 6144 nfrecs 6172 nffrec 6261 nfixpxy 6579 nfixp1 6580 xpcomco 6688 nfsup 6847 nfinf 6872 nfdju 6895 caucvgprprlemaddq 7484 nfseq 10196 nfsum1 11093 nfsum 11094 |
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