Theorem equvinva 2057

Description: A modified version of the forward implication of equvinv 2056 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 2042	. 2 ⊢ ∃𝑧 𝑦 = 𝑧
2		equtr 2048	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧))
3	2	ancrd 560	. . 3 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
4	3	eximdv 1944	. 2 ⊢ (𝑥 = 𝑦 → (∃𝑧 𝑦 = 𝑧 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧)))
5	1, 4	mpi 21	1 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑦 = 𝑧))

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807

This theorem is referenced by:  sbequ2  2291  ax13lem1  2412  nfeqf  2419  wl-ax13lem1  38065