Theorem ax17el 924

Description: Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 925 considered as a metatheorem. Do not use it for later proofs - use ax-17 925 instead, to avoid reference to the redundant ax-15 806.)

Assertion

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

Distinct variable group(s):   x,z   y,z

Proof of Theorem ax17el

Step	Hyp	Ref	Expression
1		ax-15 806	. 2 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x ∈ y → ∀z x ∈ y)))
2		ax-16 922	. 2 ⊢ (∀z z = x → (x ∈ y → ∀z x ∈ y))
3		ax-16 922	. 2 ⊢ (∀z z = y → (x ∈ y → ∀z x ∈ y))
4	1, 2, 3	pm2.61ii 113	1 ⊢ (x ∈ y → ∀z x ∈ y)

Syntax hints:   → wi 2  ∀wal 672   = weq 797   ∈ wel 803

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-15 806  ax-16 922

metamath.org