Theorem equvelvOLD 2133

Description: Obsolete version of equvelv 2132 as of 12-Jul-2022. (Contributed by Wolf Lammen, 10-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvelvOLD

Step	Hyp	Ref	Expression
1		equtrr 2121	. . 3 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦))
2	1	alrimiv 2023	. 2 ⊢ (𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦))
3		equs4v 2102	. . 3 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) → ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))
4		equvinv 2129	. . 3 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))
5	3, 4	sylibr 226	. 2 ⊢ (∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦) → 𝑥 = 𝑦)
6	2, 5	impbii 201	1 ⊢ (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥 → 𝑧 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385  ∀wal 1651  ∃wex 1875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107

This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876

This theorem is referenced by: (None)