Theorem ax17el 2189

Description: Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-17 1616 considered as a metatheorem.) (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax17el	⊢ (x ∈ y → ∀z x ∈ y)

Distinct variable groups:   x,z   y,z

Proof of Theorem ax17el

Step	Hyp	Ref	Expression
1		ax-15 2143	. 2 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x ∈ y → ∀z x ∈ y)))
2		ax-16 2144	. 2 ⊢ (∀z z = x → (x ∈ y → ∀z x ∈ y))
3		ax-16 2144	. 2 ⊢ (∀z z = y → (x ∈ y → ∀z x ∈ y))
4	1, 2, 3	pm2.61ii 157	1 ⊢ (x ∈ y → ∀z x ∈ y)

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-15 2143  ax-16 2144

This theorem is referenced by:  dveel2ALT  2191