This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ∃ 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 )