Theorem a16g-o 2186

Description: A generalization of Axiom ax-16 2144. Version of a16g 1945 using ax-10o 2139. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
a16g-o	⊢ (∀x x = y → (φ → ∀zφ))

Distinct variable group:   x,y

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

Proof of Theorem a16g-o

Step	Hyp	Ref	Expression
1		aev-o 2182	. 2 ⊢ (∀x x = y → ∀z z = x)
2		ax-16 2144	. 2 ⊢ (∀x x = y → (φ → ∀xφ))
3		biidd 228	. . . 4 ⊢ (∀z z = x → (φ ↔ φ))
4	3	dral1-o 2154	. . 3 ⊢ (∀z z = x → (∀zφ ↔ ∀xφ))
5	4	biimprd 214	. 2 ⊢ (∀z z = x → (∀xφ → ∀zφ))
6	1, 2, 5	sylsyld 52	1 ⊢ (∀x x = y → (φ → ∀zφ))

Syntax hints:   → wi 4  ∀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  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142  ax-16 2144

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:  ax11inda2  2199