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Mirrors > Home > MPE Home > Th. List > Mathboxes > naecoms-o | Structured version Visualization version GIF version |
Description: A commutation rule for distinct variable specifiers. Version of naecoms 2313 using ax-c11 34172. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nalequcoms-o.1 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑) |
Ref | Expression |
---|---|
naecoms-o | ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aecom-o 34186 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | |
2 | nalequcoms-o.1 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑) | |
3 | 1, 2 | nsyl4 156 | . 2 ⊢ (¬ 𝜑 → ∀𝑦 𝑦 = 𝑥) |
4 | 3 | con1i 144 | 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-c5 34168 ax-c4 34169 ax-c7 34170 ax-c10 34171 ax-c11 34172 ax-c9 34175 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: ax12inda2ALT 34231 |
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