Theorem 2ax6e 2448

Description: We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2447 with a distinct variable constraint. (Contributed by Wolf Lammen, 28-Sep-2018.)

Assertion

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

Distinct variable group:   𝑧,𝑤

Proof of Theorem 2ax6e

Step	Hyp	Ref	Expression
1		aeveq 1980	. . . 4 ⊢ (∀𝑤 𝑤 = 𝑧 → 𝑧 = 𝑥)
2		aeveq 1980	. . . 4 ⊢ (∀𝑤 𝑤 = 𝑧 → 𝑤 = 𝑦)
3	1, 2	jca 554	. . 3 ⊢ (∀𝑤 𝑤 = 𝑧 → (𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
4		19.8a 2050	. . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
5		19.8a 2050	. . 3 ⊢ (∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
6	3, 4, 5	3syl 18	. 2 ⊢ (∀𝑤 𝑤 = 𝑧 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
7		2ax6elem 2447	. 2 ⊢ (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
8	6, 7	pm2.61i 176	1 ⊢ ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)

Syntax hints:   ∧ wa 384  ∀wal 1479  ∃wex 1702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708

This theorem is referenced by:  2sb5rf  2449  2sb6rf  2450