MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2euswap Structured version   Visualization version   GIF version

Theorem 2euswap 2686
Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
Assertion
Ref Expression
2euswap (∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))

Proof of Theorem 2euswap
StepHypRef Expression
1 excomim 2192 . . . 4 (∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)
21a1i 11 . . 3 (∀𝑥∃*𝑦𝜑 → (∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑))
3 2moswap 2685 . . 3 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))
42, 3anim12d 587 . 2 (∀𝑥∃*𝑦𝜑 → ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) → (∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑)))
5 eu5 2633 . 2 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑))
6 eu5 2633 . 2 (∃!𝑦𝑥𝜑 ↔ (∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑))
74, 5, 63imtr4g 285 1 (∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1630  wex 1853  ∃!weu 2607  ∃*wmo 2608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-eu 2611  df-mo 2612
This theorem is referenced by:  2eu1  2691  euxfr2  3532  2reuswap  3551  2reuswap2  29657
  Copyright terms: Public domain W3C validator