Theorem elequ2g 2160

Description: A form of elequ2 2159 with a universal quantifier. Its converse is the axiom of extensionality ax-ext 2732. (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 2159	. 2 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
2	1	alrimiv 1959	1 ⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  axextb  2735