Theorem 2eu2 2678

Description: Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2402. (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 2604	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 → ∃*𝑦∃𝑥𝜑)
2		2moex 2666	. . 3 ⊢ (∃*𝑦∃𝑥𝜑 → ∀𝑥∃*𝑦𝜑)
3		2eu1 2676	. . . 4 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑)))
4		simpl 486	. . . 4 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃𝑦𝜑)
5	3, 4	biimtrdi 255	. . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))
6	1, 2, 5	3syl 18	. 2 ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥∃𝑦𝜑))
7		2exeu 2672	. . 3 ⊢ ((∃!𝑥∃𝑦𝜑 ∧ ∃!𝑦∃𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
8	7	expcom 417	. 2 ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑥∃!𝑦𝜑))
9	6, 8	impbid 214	1 ⊢ (∃!𝑦∃𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥∃𝑦𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557  ∃wex 1798  ∃*wmo 2563  ∃!weu 2594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-11 2190  ax-12 2211  ax-13 2402

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-mo 2565  df-eu 2595

This theorem is referenced by:  2eu8  2684