Theorem axc16b 5007

Description: This theorem shows that axiom ax-c16 34681 is redundant in the presence of theorem dtru 5006, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness (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 5006	. 2 ⊢ ¬ ∀𝑥 𝑥 = 𝑦
2	1	pm2.21i 116	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-nul 4941  ax-pow 4992

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859

This theorem is referenced by: (None)