Theorem bj-elequ2g 31853

Description: A form of elequ2 1991 with a universal quantifier. Its converse is ax-ext 2590. (TODO: move to main part, minimize axext4 2594--- as of 4-Nov-2020, minimizes only axext4 2594, by 13 bytes; and link to it in the comment of ax-ext 2590.) (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 1991	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
2	1	alrimiv 1842	1 ⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by:  bj-axext4  31958  bj-cleqhyp  32084