Metamath Proof Explorer


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 𝑧𝑤 ( 𝑧 = 𝑥𝑤 = 𝑦 )

Proof

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