Theorem 2eu2 2541

Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)

Assertion

Ref	Expression
2eu2	⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥∃𝑦𝜑))

Proof of Theorem 2eu2

Step	Hyp	Ref	Expression
1		eumo 2486	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∃*𝑦∃𝑥𝜑)
2		2moex 2530	. . 3 ⊢ (∃*𝑦∃𝑥𝜑 → ∀𝑥∃*𝑦𝜑)
3		2eu1 2540	. . . 4 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
4		simpl 471	. . . 4 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃𝑦𝜑)
5	3, 4	syl6bi 241	. . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))
6	1, 2, 5	3syl 18	. 2 ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))
7		2exeu 2536	. . 3 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
8	7	expcom 449	. 2 ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑥∃!𝑦𝜑))
9	6, 8	impbid 200	1 ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥∃𝑦𝜑))

Syntax hints:   → wi 4   ↔ wb 194   ∧ wa 382  ∀wal 1472  ∃wex 1694  ∃!weu 2457  ∃*wmo 2458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-eu 2461  df-mo 2462

This theorem is referenced by:  2eu8  2547