Theorem 2euswapv 2624

Description: A condition allowing to swap an existential quantifier and a unique existential quantifier. Version of 2euswap 2639 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 10-Apr-2004.) (Revised by GG, 22-Aug-2023.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2euswapv

Step	Hyp	Ref	Expression
1		excomim 2165	. . . 4 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)
2	1	a1i 11	. . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑))
3		2moswapv 2623	. . 3 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))
4	2, 3	anim12d 609	. 2 ⊢ (∀𝑥∃*𝑦𝜑 → ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑)))
5		df-eu 2563	. 2 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑))
6		df-eu 2563	. 2 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑))
7	4, 5, 6	3imtr4g 296	1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539  ∃wex 1780  ∃*wmo 2532  ∃!weu 2562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2143  ax-11 2159  ax-12 2179

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-mo 2534  df-eu 2563

This theorem is referenced by:  2eu1v  2646  euxfr2w  3677  2reuswap  3703  2reuswap2  3704  reuxfrdf  32460