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Theorem axc16b 5277
 Description: This theorem shows that axiom ax-c16 36098 is redundant in the presence of theorem dtru 5258, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5258 (which links to this theorem). (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 7-Nov-2006.)
Assertion
Ref Expression
axc16b (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem axc16b
StepHypRef Expression
1 dtru 5258 . 2 ¬ ∀𝑥 𝑥 = 𝑦
21pm2.21i 119 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-nul 5196  ax-pow 5253 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-nf 1786 This theorem is referenced by: (None)
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