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Mirrors > Home > MPE Home > Th. List > nfal | Structured version Visualization version GIF version |
Description: If 𝑥 is not free in 𝜑, then it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfal.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfal | ⊢ Ⅎ𝑥∀𝑦𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfal.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nf5ri 2192 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | 2 | hbal 2164 | . 2 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) |
4 | 3 | nf5i 2143 | 1 ⊢ Ⅎ𝑥∀𝑦𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1534 Ⅎwnf 1779 |
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-10 2138 ax-11 2154 ax-12 2174 |
This theorem depends on definitions: df-bi 207 df-ex 1776 df-nf 1780 |
This theorem is referenced by: nfex 2322 nfnf 2324 nfsbvOLD 2329 aaanOLD 2332 cbval2v 2343 pm11.53 2346 19.12vv 2347 cbval2 2413 nfsb4t 2501 mof 2560 euf 2573 2eu3 2651 axextmo 2709 nfnfc1 2905 nfnfc 2915 sbcnestgfw 4426 sbcnestgf 4431 nfdisjw 5126 nfdisj 5127 nfdisj1 5128 axrep1 5285 axrep2 5287 axrep3 5288 axrep4OLD 5291 nffr 5661 zfcndrep 10651 zfcndinf 10655 mreexexd 17692 mptelee 28924 mo5f 32516 iinabrex 32588 19.12b 35782 bj-cbv2v 36780 ax11-pm2 36818 wl-sb8t 37532 wl-mo2tf 37551 wl-eutf 37553 wl-mo2t 37555 wl-mo3t 37556 wl-sb8eut 37558 wl-sb8eutv 37559 mpobi123f 38148 pm11.57 44384 pm11.59 44386 ichnfimlem 47387 ichnfim 47388 nfsetrecs 48916 pgind 48947 |
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