Theorem a16gb 934

Description: A generalization of axiom ax-16 922.

Assertion

Ref	Expression
a16gb	⊢ (∀x x = y → (φ ↔ ∀zφ))

Distinct variable group(s):   x,y

Proof of Theorem a16gb

Step	Hyp	Ref	Expression
1		a16g 933	. 2 ⊢ (∀x x = y → (φ → ∀zφ))
2		ax-4 673	. . 3 ⊢ (∀zφ → φ)
3	2	a1i 7	. 2 ⊢ (∀x x = y → (∀zφ → φ))
4	1, 3	impbid 397	1 ⊢ (∀x x = y → (φ ↔ ∀zφ))

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

This theorem is referenced by:  sbal 997

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  ax-16 922

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

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