Theorem ax16ALT 1275

Description: Version of ax16 1213 that doesn't require ax-10 970 or ax-12 972 for its proof.

Assertion

Ref	Expression
ax16ALT	⊢ (∀x x = y → (φ → ∀xφ))

Distinct variable group:   x,y

Proof of Theorem ax16ALT

Step	Hyp	Ref	Expression
1		sbequ12 1185	. 2 ⊢ (x = z → (φ ↔ [z / x]φ))
2		ax-17 975	. . 3 ⊢ (φ → ∀zφ)
3	2	hbsb3 1210	. 2 ⊢ ([z / x]φ → ∀x[z / x]φ)
4	1, 3	ax16i 1274	1 ⊢ (∀x x = y → (φ → ∀xφ))

Syntax hints:   → wi 3  ∀wal 958   = wceq 960  [wsbc 1174

This theorem is referenced by:  dvelimALT 1357

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-11 971  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1176

Copyright terms: Public domain