Theorem equvinva 2031

Description: A modified version of the forward implication of equvinv 2030 adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018.)

Assertion

Ref	Expression
equvinva	⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinva

Step	Hyp	Ref	Expression
1		ax6evr 2016	. 2 ⊢ ∃𝑧 𝑦 = 𝑧
2		equtr 2022	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧))
3	2	ancrd 551	. . 3 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
4	3	eximdv 1918	. 2 ⊢ (𝑥 = 𝑦 → (∃𝑧 𝑦 = 𝑧 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
5	1, 4	mpi 20	1 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧))

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1780

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 1968  ax-7 2009

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781

This theorem is referenced by:  sbequ2  2252  ax13lem1  2374  nfeqf  2381  wl-ax13lem1  37538