Theorem eq4ds 823

Description: A commutation rule for distinct variable specifiers.

Hypothesis

Ref	Expression
eq4ds.1	⊢ (¬ ∀x x = y → φ)

Assertion

Ref	Expression
eq4ds	⊢ (¬ ∀y y = x → φ)

Proof of Theorem eq4ds

Step	Hyp	Ref	Expression
1		eq4 821	. . 3 ⊢ (∀x x = y → ∀y y = x)
2		eq4ds.1	. . 3 ⊢ (¬ ∀x x = y → φ)
3	1, 2	nsyl4 105	. 2 ⊢ (¬ φ → ∀y y = x)
4	3	con1i 88	1 ⊢ (¬ ∀y y = x → φ)

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

This theorem is referenced by:  sbcom 916  ralcom2 1314  nd5 3736  axrepndlem1 3738  axrepndlem2 3739  axrepnd 3740  axpowndlem3 3745  axpownd 3747

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-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-12 802

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

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