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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 2372. See nfcv 2904 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 2933 | . 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥) | |
2 | 1 | naecoms 2429 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 Ⅎwnfc 2884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-nfc 2886 |
This theorem is referenced by: dfid3 5578 oprabid 7441 axrepndlem1 10587 axrepndlem2 10588 axrepnd 10589 axunnd 10591 axpowndlem3 10594 axpowndlem4 10595 axpownd 10596 axregndlem2 10598 axinfndlem1 10600 axinfnd 10601 axacndlem4 10605 axacndlem5 10606 axacnd 10607 bj-nfcsym 35779 |
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