Theorem equvinva 2063

Description: A modified version of the forward implication of equvinv 2062 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 2048	. 2 ⊢ ∃𝑧 𝑦 = 𝑧
2		equtr 2054	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧))
3	2	ancrd 561	. . 3 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
4	3	eximdv 1950	. 2 ⊢ (𝑥 = 𝑦 → (∃𝑧 𝑦 = 𝑧 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
5	1, 4	mpi 21	1 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  sbequ2  2286  ax13lem1  2405  nfeqf  2412  wl-ax13lem1  38233