Theorem equvelv 2061

Description: A biconditional form of equvel 2488 with disjoint variable conditions and proved from Tarski's FOL axiom schemes. (Contributed by Andrew Salmon, 2-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Apr-2021.) (Proof shortened by Wolf Lammen, 12-Jul-2022.)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvelv

Step	Hyp	Ref	Expression
1		equequ1 2055	. 2 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 ↔ 𝑥 = 𝑦))
2	1	equsalvw 2034	1 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) ↔ 𝑥 = 𝑦)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is used by:  eu6lem  2601