Theorem 2euswap 2666

Description: A condition allowing to swap an existential quantifier and a unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2397. Use the weaker 2euswapv 2651 when possible. (Contributed by NM, 10-Apr-2004.) (New usage is discouraged.)

Assertion

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

Proof of Theorem 2euswap

Step	Hyp	Ref	Expression
1		excomim 2191	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
2	1	a1i 11	. . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑))
3		2moswap 2665	. . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))
4	2, 3	anim12d 617	. 2 ⊢ (∀𝑥∃*𝑦𝜑 → ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑)))
5		df-eu 2590	. 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑))
6		df-eu 2590	. 2 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑))
7	4, 5, 6	3imtr4g 298	1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1552  ∃wex 1793  ∃*wmo 2558  ∃!weu 2589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-10 2169  ax-11 2185  ax-12 2206  ax-13 2397

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1557  df-ex 1794  df-nf 1798  df-mo 2560  df-eu 2590

This theorem is referenced by:  2eu1  2671  euxfr2  3679