Theorem ddeeq2 1002

Description: Quantifier introduction when one pair of variables is distinct.

Assertion

Ref	Expression
ddeeq2	⊢ (¬ ∀x x = y → (z = y → ∀x z = y))

Distinct variable group(s):   x,z

Proof of Theorem ddeeq2

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (z = w → ∀x z = w)
2		eqt2b 818	. 2 ⊢ (w = y → (z = w ↔ z = y))
3	1, 2	ddelim 1000	1 ⊢ (¬ ∀x x = y → (z = y → ∀x z = y))

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672   = weq 797

This theorem is referenced by:  nd5 3736  axrepndlem1 3738  axpowndlem2 3744  axpowndlem3 3745  axacndlem5 3757

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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