Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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. (Contributed by Mario Carneiro, 5-Dec-2016.) |
Ref | Expression |
---|---|
nfcvf2 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcvf 2788 | . 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥) | |
2 | 1 | naecoms 2313 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 Ⅎwnfc 2751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-cleq 2615 df-clel 2618 df-nfc 2753 |
This theorem is referenced by: dfid3 5025 oprabid 6677 axrepndlem1 9414 axrepndlem2 9415 axrepnd 9416 axunnd 9418 axpowndlem3 9421 axpowndlem4 9422 axpownd 9423 axregndlem2 9425 axinfndlem1 9427 axinfnd 9428 axacndlem4 9432 axacndlem5 9433 axacnd 9434 bj-nfcsym 32886 |
Copyright terms: Public domain | W3C validator |