Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > naecoms | Structured version Visualization version GIF version |
Description: A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.) |
Ref | Expression |
---|---|
naecoms.1 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑) |
Ref | Expression |
---|---|
naecoms | ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aecom 2311 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥) | |
2 | naecoms.1 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑) | |
3 | 1, 2 | sylnbir 321 | 1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 |
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-10 2019 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 |
This theorem is referenced by: sb9 2426 eujustALT 2473 nfcvf2 2789 axpowndlem2 9420 wl-sbcom2d 33344 wl-mo2df 33352 wl-eudf 33354 |
Copyright terms: Public domain | W3C validator |