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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 2390. See nfcv 2979 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 3009 | . 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥) | |
2 | 1 | naecoms 2451 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1535 Ⅎwnfc 2963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-nfc 2965 |
This theorem is referenced by: dfid3 5464 oprabid 7190 axrepndlem1 10016 axrepndlem2 10017 axrepnd 10018 axunnd 10020 axpowndlem3 10023 axpowndlem4 10024 axpownd 10025 axregndlem2 10027 axinfndlem1 10029 axinfnd 10030 axacndlem4 10034 axacndlem5 10035 axacnd 10036 bj-nfcsym 34217 |
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