Theorem equvinivOLD 1948

Description: The forward implication of equvinv 1946. Obsolete as of 11-Apr-2021. Use equvinv 1946 instead. (Contributed by Wolf Lammen, 11-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinivOLD

Step	Hyp	Ref	Expression
1		ax6ev 1877	. 2 ⊢ ∃𝑧 𝑧 = 𝑥
2		equtrr 1936	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → 𝑧 = 𝑦))
3	2	ancld 574	. . 3 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑥 → (𝑧 = 𝑥 ∧ 𝑧 = 𝑦)))
4	3	eximdv 1833	. 2 ⊢ (𝑥 = 𝑦 → (∃𝑧 𝑧 = 𝑥 → ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦)))
5	1, 4	mpi 20	1 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696

This theorem is referenced by: (None)