Theorem ddeel1 1003

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

Assertion

Ref	Expression
ddeel1	⊢ (¬ ∀x x = y → (y ∈ z → ∀x y ∈ z))

Distinct variable group(s):   x,z

Proof of Theorem ddeel1

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (w ∈ z → ∀x w ∈ z)
2		a13b 819	. 2 ⊢ (w = y → (w ∈ z ↔ y ∈ z))
3	1, 2	ddelim 1000	1 ⊢ (¬ ∀x x = y → (y ∈ z → ∀x y ∈ z))

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672   = weq 797   ∈ wel 803

This theorem is referenced by:  axrepndlem2 3739  axunnd 3742  axpowndlem2 3744  axpowndlem3 3745  axpowndlem4 3746  axpownd 3747  axregndlem2 3749  axinfndlem1 3751  axacndlem4 3756  axacnd 3758

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-13 804  ax-17 925

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

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