Theorem 2ax6e 2471

Description: We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2470 with a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by Wolf Lammen, 28-Sep-2018.) (Proof shortened by Wolf Lammen, 3-Oct-2023.) (New usage is discouraged.)

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 2060	. . . . 5 ⊢ (∀𝑤 𝑤 = 𝑧 → 𝑧 = 𝑥)
2		aeveq 2060	. . . . 5 ⊢ (∀𝑤 𝑤 = 𝑧 → 𝑤 = 𝑦)
3	1, 2	jca 511	. . . 4 ⊢ (∀𝑤 𝑤 = 𝑧 → (𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
4	3	19.8ad 2177	. . 3 ⊢ (∀𝑤 𝑤 = 𝑧 → ∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
5	4	19.8ad 2177	. 2 ⊢ (∀𝑤 𝑤 = 𝑧 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
6		2ax6elem 2470	. 2 ⊢ (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦))
7	5, 6	pm2.61i 182	1 ⊢ ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)

Syntax hints:   ∧ wa 395  ∀wal 1537  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788

This theorem is referenced by:  2sb5rf  2472  2sb6rf  2473