Theorem 2ax6e

Description: We can always find values matching x and y , as long as they are represented by distinct variables. Version of 2ax6elem with a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Wolf Lammen, 28-Sep-2018) (Proof shortened by Wolf Lammen, 3-Oct-2023) (New usage is discouraged.)

	Ref	Expression
Assertion	2ax6e	⊢ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 )

Proof

Step	Hyp	Ref	Expression
1		aeveq	⊢ ( ∀ 𝑤 𝑤 = 𝑧 → 𝑧 = 𝑥 )
2		aeveq	⊢ ( ∀ 𝑤 𝑤 = 𝑧 → 𝑤 = 𝑦 )
3	1 2	jca	⊢ ( ∀ 𝑤 𝑤 = 𝑧 → ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) )
4	3	19.8ad	⊢ ( ∀ 𝑤 𝑤 = 𝑧 → ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) )
5	4	19.8ad	⊢ ( ∀ 𝑤 𝑤 = 𝑧 → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) )
6		2ax6elem	⊢ ( ¬ ∀ 𝑤 𝑤 = 𝑧 → ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 ) )
7	5 6	pm2.61i	⊢ ∃ 𝑧 ∃ 𝑤 ( 𝑧 = 𝑥 ∧ 𝑤 = 𝑦 )