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Metamath Proof Explorer

Theorem axc16b

Description: This theorem shows that Axiom ax-c16 is redundant in the presence of Theorem dtruALT2 , 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)

Ref Expression
Assertion axc16b ⊢ ∀ x x = y → φ → ∀ x φ

Proof

Step Hyp Ref Expression
1 dtruALT2 ⊢ ¬ ∀ x x = y
2 1 pm2.21i ⊢ ∀ x x = y → φ → ∀ x φ