Theorem ax5el 36951

Description: Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-5 1913 considered as a metatheorem.) (Contributed by NM, 22-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax5el	⊢ (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem ax5el

Step	Hyp	Ref	Expression
1		ax-c14 36905	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)))
2		ax-c16 36906	. 2 ⊢ (∀𝑧 𝑧 = 𝑥 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))
3		ax-c16 36906	. 2 ⊢ (∀𝑧 𝑧 = 𝑦 → (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦))
4	1, 2, 3	pm2.61ii 183	1 ⊢ (𝑥 ∈ 𝑦 → ∀𝑧 𝑥 ∈ 𝑦)

Syntax hints:   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-c14 36905  ax-c16 36906

This theorem is referenced by:  dveel2ALT  36953