Theorem 2eu2 2674

Description: Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.)

Assertion

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

Proof of Theorem 2eu2

Step	Hyp	Ref	Expression
1		eumo 2598	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∃*𝑦∃𝑥𝜑)
2		2moex 2662	. . 3 ⊢ (∃*𝑦∃𝑥𝜑 → ∀𝑥∃*𝑦𝜑)
3		2eu1 2672	. . . 4 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
4		simpl 487	. . . 4 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃𝑦𝜑)
5	3, 4	syl6bi 256	. . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))
6	1, 2, 5	3syl 18	. 2 ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))
7		2exeu 2668	. . 3 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
8	7	expcom 418	. 2 ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑥∃!𝑦𝜑))
9	6, 8	impbid 215	1 ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥∃𝑦𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1537  ∃wex 1782  ∃*wmo 2556  ∃!weu 2588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2380

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589

This theorem is referenced by:  2eu8  2681