Theorem equsb1vOLDOLD 2517

Description: Obsolete version of equsb1v 2112 as of 19-Jun-2023. (Contributed by BJ, 11-Sep-2019.) Remove dependencies on axioms. (Revised by Wolf Lammen, 30-May-2023.) (Proof shortened by Steven Nguyen, 31-May-2023.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
equsb1vOLDOLD	⊢ [𝑦 / 𝑥]𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem equsb1vOLDOLD

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
2		ax6ev 1972	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3	1	ancli 551	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑦 ∧ 𝑥 = 𝑦))
4	2, 3	eximii 1837	. 2 ⊢ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑦)
5		dfsb1 2510	. 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑦 ↔ ((𝑥 = 𝑦 → 𝑥 = 𝑦) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 = 𝑦)))
6	1, 4, 5	mpbir2an 709	1 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1780  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by: (None)