Theorem del35 836

Description: A distinctor elimination lemma. Formula-builder for universal quantifier.

Hypothesis

Ref	Expression
del35.1	⊢ (∀x x = y → (φ → ψ))

Assertion

Ref	Expression
del35	⊢ (∀x x = y → (∀yφ → ∀xψ))

Proof of Theorem del35

Step	Hyp	Ref	Expression
1		del35.1	. . . 4 ⊢ (∀x x = y → (φ → ψ))
2	1	19.20ii 692	. . 3 ⊢ (∀y∀x x = y → (∀yφ → ∀yψ))
3	2	eq5s 825	. 2 ⊢ (∀x x = y → (∀yφ → ∀yψ))
4		ax-10 800	. . 3 ⊢ (∀y y = x → (∀yψ → ∀xψ))
5	4	eq4s 822	. 2 ⊢ (∀x x = y → (∀yψ → ∀xψ))
6	3, 5	syld 27	1 ⊢ (∀x x = y → (∀yφ → ∀xψ))

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

This theorem is referenced by:  del34b 837  del40 839  hbsb4 905  sb9i 920  a16g 933

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-12 802

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

metamath.org