Theorem elequ2g 2124

Description: A form of elequ2 2123 with a universal quantifier. Its converse is the axiom of extensionality ax-ext 2711. (Contributed by BJ, 3-Oct-2019.)

Assertion

Ref	Expression
elequ2g	⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem elequ2g

Step	Hyp	Ref	Expression
1		elequ2 2123	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
2	1	alrimiv 1926	1 ⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778

This theorem is referenced by:  axextb  2714