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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, 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 2053 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | 2 | hbal 2023 | . 2 ⊢ (∀𝑦𝜑 → ∀𝑥∀𝑦𝜑) |
4 | 3 | nf5i 2011 | 1 ⊢ Ⅎ𝑥∀𝑦𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1473 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 df-nf 1701 |
This theorem is referenced by: nfex 2140 nfnf 2144 nfaldOLD 2152 nfa2OLD 2154 aaan 2156 cbv3v 2158 pm11.53 2167 19.12vv 2168 cbv3 2253 cbvalv 2261 cbval2 2267 nfsb4t 2377 euf 2466 mo2 2467 2eu3 2543 bm1.1 2595 nfnfc1 2754 nfnfc 2760 nfnfcALT 2761 sbcnestgf 3947 dfnfc2OLD 4391 nfdisj 4565 nfdisj1 4566 axrep1 4700 axrep2 4701 axrep3 4702 axrep4 4703 nffr 5012 zfcndrep 9315 zfcndinf 9319 mreexexd 16131 mreexexdOLD 16132 mptelee 25575 mo5f 28708 19.12b 30951 bj-cbv2v 31919 bj-cbval2v 31924 bj-axrep1 31976 bj-axrep2 31977 bj-axrep3 31978 bj-axrep4 31979 ax11-pm2 32011 bj-nfnfc 32047 wl-sb8t 32512 wl-mo2tf 32532 wl-eutf 32534 wl-mo2t 32536 wl-mo3t 32537 wl-sb8eut 32538 mpt2bi123f 33141 pm11.57 37611 pm11.59 37613 nfsetrecs 42232 |
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