Theorem axc16b 5350

Description: This theorem shows that Axiom ax-c16 40042 is redundant in the presence of Theorem dtruALT2 5331, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness 5331 (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		dtruALT2 5331	. 2 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2	1	pm2.21i 120	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-nul 5259  ax-pow 5326

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by: (None)