Theorem euequOLD 2618

Description: Obsolete proof of euequ 2617 as of 28-Feb-2023. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof shortened by Wolf Lammen, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
euequOLD	⊢ ∃!𝑥 𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem euequOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax6evr 1973	. . 3 ⊢ ∃𝑧 𝑦 = 𝑧
2		equequ2 1984	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
3	2	alrimiv 1887	. . 3 ⊢ (𝑦 = 𝑧 → ∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
4	1, 3	eximii 1800	. 2 ⊢ ∃𝑧∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧)
5		eu6 2591	. 2 ⊢ (∃!𝑥 𝑥 = 𝑦 ↔ ∃𝑧∀𝑥(𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
6	4, 5	mpbir 223	1 ⊢ ∃!𝑥 𝑥 = 𝑦

Syntax hints:   ↔ wb 198  ∀wal 1506  ∃wex 1743  ∃!weu 2584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-10 2080  ax-12 2107

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-ex 1744  df-nf 1748  df-mo 2548  df-eu 2585

This theorem is referenced by: (None)