Metamath Proof Explorer


Theorem nd1

Description: A lemma for proving conditionless ZFC axioms. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 1-Jan-2002) (New usage is discouraged.)

Ref Expression
Assertion nd1 ( ∀ 𝑥 𝑥 = 𝑦 → ¬ ∀ 𝑥 𝑦𝑧 )

Proof

Step Hyp Ref Expression
1 elirrv ¬ 𝑧𝑧
2 stdpc4 ( ∀ 𝑦 𝑦𝑧 → [ 𝑧 / 𝑦 ] 𝑦𝑧 )
3 1 nfnth 𝑦 𝑧𝑧
4 elequ1 ( 𝑦 = 𝑧 → ( 𝑦𝑧𝑧𝑧 ) )
5 3 4 sbie ( [ 𝑧 / 𝑦 ] 𝑦𝑧𝑧𝑧 )
6 2 5 sylib ( ∀ 𝑦 𝑦𝑧𝑧𝑧 )
7 1 6 mto ¬ ∀ 𝑦 𝑦𝑧
8 axc11 ( ∀ 𝑥 𝑥 = 𝑦 → ( ∀ 𝑥 𝑦𝑧 → ∀ 𝑦 𝑦𝑧 ) )
9 7 8 mtoi ( ∀ 𝑥 𝑥 = 𝑦 → ¬ ∀ 𝑥 𝑦𝑧 )