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Theorem naecoms-o 33230
 Description: A commutation rule for distinct variable specifiers. Version of naecoms 2301 using ax-c11 33190. (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 33204 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 nalequcoms-o.1 . . 3 (¬ ∀𝑥 𝑥 = 𝑦𝜑)
31, 2nsyl4 155 . 2 𝜑 → ∀𝑦 𝑦 = 𝑥)
43con1i 143 1 (¬ ∀𝑦 𝑦 = 𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-c5 33186  ax-c4 33187  ax-c7 33188  ax-c10 33189  ax-c11 33190  ax-c9 33193 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by:  ax12inda2ALT  33249
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