Theorem ax17el 1359

Description: Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-17 969 considered as a metatheorem.)

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 1358	. 2 ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x ∈ y → ∀z x ∈ y)))
2		ax-16 1208	. 2 ⊢ (∀z z = x → (x ∈ y → ∀z x ∈ y))
3		ax-16 1208	. 2 ⊢ (∀z z = y → (x ∈ y → ∀z x ∈ y))
4	1, 2, 3	pm2.61ii 130	1 ⊢ (x ∈ y → ∀z x ∈ y)

Syntax hints:   → wi 3  ∀wal 952   ∈ wcel 956

This theorem is referenced by:  dveel2ALT 1360

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-16 1208  ax-15 1358

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