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Theorem naecoms-o 33731
Description: A commutation rule for distinct variable specifiers. Version of naecoms 2312 using ax-c11 33691. (Contributed by NM, 2-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
nalequcoms-o.1 (¬ ∀𝑥 𝑥 = 𝑦𝜑)
Assertion
Ref Expression
naecoms-o (¬ ∀𝑦 𝑦 = 𝑥𝜑)

Proof of Theorem naecoms-o
StepHypRef Expression
1 aecom-o 33705 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 nalequcoms-o.1 . . 3 (¬ ∀𝑥 𝑥 = 𝑦𝜑)
31, 2nsyl4 156 . 2 𝜑 → ∀𝑦 𝑦 = 𝑥)
43con1i 144 1 (¬ ∀𝑦 𝑦 = 𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-c5 33687  ax-c4 33688  ax-c7 33689  ax-c10 33690  ax-c11 33691  ax-c9 33694
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by:  ax12inda2ALT  33750
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