Theorem del34b 837

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

Hypothesis

Ref	Expression
del34b.1	⊢ (∀x x = y → (φ ↔ ψ))

Assertion

Ref	Expression
del34b	⊢ (∀x x = y → (∀xφ ↔ ∀yψ))

Proof of Theorem del34b

Step	Hyp	Ref	Expression
1		del34b.1	. . . 4 ⊢ (∀x x = y → (φ ↔ ψ))
2	1	biimpd 135	. . 3 ⊢ (∀x x = y → (φ → ψ))
3	2	del34 835	. 2 ⊢ (∀x x = y → (∀xφ → ∀yψ))
4	1	biimprd 136	. . 3 ⊢ (∀x x = y → (ψ → φ))
5	4	del35 836	. 2 ⊢ (∀x x = y → (∀yψ → ∀xφ))
6	3, 5	impbid 397	1 ⊢ (∀x x = y → (∀xφ ↔ ∀yψ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = weq 797

This theorem is referenced by:  ralcom2 1314  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-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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