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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 2426 using ax-c11 36595. (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 36609 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | |
2 | nalequcoms-o.1 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → 𝜑) | |
3 | 1, 2 | nsyl4 161 | . 2 ⊢ (¬ 𝜑 → ∀𝑦 𝑦 = 𝑥) |
4 | 3 | con1i 149 | 1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-c5 36591 ax-c4 36592 ax-c7 36593 ax-c10 36594 ax-c11 36595 ax-c9 36598 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 |
This theorem is referenced by: ax12inda2ALT 36654 |
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