Theorem a16gb 2050

Description: A generalization of Axiom ax-16 2144. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem a16gb

Step	Hyp	Ref	Expression
1		a16g 1945	. 2 ⊢ (∀x x = y → (φ → ∀zφ))
2		sp 1747	. 2 ⊢ (∀zφ → φ)
3	1, 2	impbid1 194	1 ⊢ (∀x x = y → (φ ↔ ∀zφ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  sbal  2127