Theorem equvelv 1950

Description: A specialized version of equvel 2335 with distinct variable restrictions and fewer axiom usage. (Contributed by Wolf Lammen, 10-Apr-2021.)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvelv

Step	Hyp	Ref	Expression
1		equtrr 1936	. . 3 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦))
2	1	alrimiv 1842	. 2 ⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦))
3		equs4v 1917	. . 3 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) → ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))
4		equvinv 1946	. . 3 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))
5	3, 4	sylibr 223	. 2 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) → 𝑥 = 𝑦)
6	2, 5	impbii 198	1 ⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473  ∃wex 1695

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

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

This theorem is referenced by: (None)