Theorem axc16b 5255

Description: This theorem shows that axiom ax-c16 36188 is redundant in the presence of theorem dtru 5236, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5236 (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

Step	Hyp	Ref	Expression
1		dtru 5236	. 2 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2	1	pm2.21i 119	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-12 2175  ax-nul 5174  ax-pow 5231

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)