Theorem equvelv 1963

Description: A specialized version of equvel 2347 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 1949	. . 3 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦))
2	1	alrimiv 1855	. 2 ⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦))
3		equs4v 1930	. . 3 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) → ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))
4		equvinv 1959	. . 3 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))
5	3, 4	sylibr 224	. 2 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) → 𝑥 = 𝑦)
6	2, 5	impbii 199	1 ⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705

This theorem is referenced by: (None)