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Mirrors > Home > MPE Home > Th. List > nfcvf2 | Structured version Visualization version GIF version |
Description: If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. Usage of this theorem is discouraged because it depends on ax-13 2379. See nfcv 2955 for a version that replaces the distinctor with a disjoint variable condition, requiring fewer axioms. (Contributed by Mario Carneiro, 5-Dec-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfcvf2 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcvf 2981 | . 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥) | |
2 | 1 | naecoms 2440 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1536 Ⅎwnfc 2936 |
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 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-nfc 2938 |
This theorem is referenced by: dfid3 5427 oprabid 7167 axrepndlem1 10003 axrepndlem2 10004 axrepnd 10005 axunnd 10007 axpowndlem3 10010 axpowndlem4 10011 axpownd 10012 axregndlem2 10014 axinfndlem1 10016 axinfnd 10017 axacndlem4 10021 axacndlem5 10022 axacnd 10023 bj-nfcsym 34339 |
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