Theorem 2euswapdc 2169

Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.)

Assertion

Ref	Expression
2euswapdc	⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)))

Proof of Theorem 2euswapdc

Step	Hyp	Ref	Expression
1		excomim 1709	. . . . 5 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
2	1	a1i 9	. . . 4 ⊢ ((DECID ∃𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑))
3		2moswapdc 2168	. . . . 5 ⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)))
4	3	imp 124	. . . 4 ⊢ ((DECID ∃𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))
5	2, 4	anim12d 335	. . 3 ⊢ ((DECID ∃𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑)))
6		eu5 2125	. . 3 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑))
7		eu5 2125	. . 3 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑))
8	5, 6, 7	3imtr4g 205	. 2 ⊢ ((DECID ∃𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))
9	8	ex 115	1 ⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)))

Syntax hints:   → wi 4   ∧ wa 104  DECID wdc 839  ∀wal 1393  ∃wex 1538  ∃!weu 2077  ∃*wmo 2078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-dc 840  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081

This theorem is referenced by:  euxfr2dc  2988  2reuswapdc  3007