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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfwsuc | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.) |
Ref | Expression |
---|---|
nfwsuc.1 | ⊢ Ⅎ𝑥𝑅 |
nfwsuc.2 | ⊢ Ⅎ𝑥𝐴 |
nfwsuc.3 | ⊢ Ⅎ𝑥𝑋 |
Ref | Expression |
---|---|
nfwsuc | ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-wsuc 34257 | . 2 ⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) | |
2 | nfwsuc.1 | . . . . 5 ⊢ Ⅎ𝑥𝑅 | |
3 | 2 | nfcnv 5832 | . . . 4 ⊢ Ⅎ𝑥◡𝑅 |
4 | nfwsuc.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
5 | nfwsuc.3 | . . . 4 ⊢ Ⅎ𝑥𝑋 | |
6 | 3, 4, 5 | nfpred 6256 | . . 3 ⊢ Ⅎ𝑥Pred(◡𝑅, 𝐴, 𝑋) |
7 | 6, 4, 2 | nfinf 9414 | . 2 ⊢ Ⅎ𝑥inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅) |
8 | 1, 7 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2885 ◡ccnv 5630 Predcpred 6250 infcinf 9373 wsuccwsuc 34255 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-xp 5637 df-cnv 5639 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-sup 9374 df-inf 9375 df-wsuc 34257 |
This theorem is referenced by: (None) |
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