Theorem del42 841

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

Hypothesis

Ref	Expression
del42.1	⊢ (∀x x = y → (φ → ψ))

Assertion

Ref	Expression
del42	⊢ (∀x x = y → (∃zφ → ∃zψ))

Proof of Theorem del42

Step	Hyp	Ref	Expression
1		del42.1	. . . . 5 ⊢ (∀x x = y → (φ → ψ))
2	1	con3d 87	. . . 4 ⊢ (∀x x = y → (¬ ψ → ¬ φ))
3	2	del36 838	. . 3 ⊢ (∀x x = y → (∀z ¬ ψ → ∀z ¬ φ))
4	3	con3d 87	. 2 ⊢ (∀x x = y → (¬ ∀z ¬ φ → ¬ ∀z ¬ ψ))
5		df-ex 679	. 2 ⊢ (∃zφ ↔ ¬ ∀z ¬ φ)
6		df-ex 679	. 2 ⊢ (∃zψ ↔ ¬ ∀z ¬ ψ)
7	4, 5, 6	3imtr4g 426	1 ⊢ (∀x x = y → (∃zφ → ∃zψ))

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672  ∃wex 678   = weq 797

This theorem is referenced by:  eqvin.l1 851  axextnd 3737  axpowndlem3 3745  axregndlem1 3748  axregnd 3750

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

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