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Theorem axc16b 5394
Description: This theorem shows that Axiom ax-c16 38873 is redundant in the presence of Theorem dtruALT2 5375, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5375 (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 dtruALT2 5375 . 2 ¬ ∀𝑥 𝑥 = 𝑦
21pm2.21i 119 1 (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-nul 5311  ax-pow 5370
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776
This theorem is referenced by: (None)
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