Theorem bj-elequ2g 31687

Description: A form of elequ2 1952 with a universal quantifier. Its converse is ax-ext 2494. (TODO: move to main part, minimize axext4 2498--- as of 4-Nov-2020, minimizes only axext4 2498, by 13 bytes; and link to it in the comment of ax-ext 2494.) (Contributed by BJ, 3-Oct-2019.)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem bj-elequ2g

Step	Hyp	Ref	Expression
1		elequ2 1952	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
2	1	alrimiv 1808	1 ⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ↔ wb 194  ∀wal 1472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947

This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695

This theorem is referenced by:  bj-axext4  31799  bj-cleqhyp  31915