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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 2441 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 ↔ ∀𝑦 𝑦 = 𝑥) | |
2 | naecoms.1 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑) | |
3 | 1, 2 | sylnbir 332 | 1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-nf 1776 |
This theorem is referenced by: sb9 2554 eujustALT 2650 nfcvf2 3005 axpowndlem2 10008 wl-sbcom2d 34678 wl-mo2df 34687 wl-eudf 34689 |
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