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

Theorem ssrmof 4063
Description: "At most one" existential quantification restricted to a subclass. (Contributed by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
ssrexf.1 𝑥𝐴
ssrexf.2 𝑥𝐵
Assertion
Ref Expression
ssrmof (𝐴𝐵 → (∃*𝑥𝐵 𝜑 → ∃*𝑥𝐴 𝜑))

Proof of Theorem ssrmof
StepHypRef Expression
1 ssrexf.1 . . . . 5 𝑥𝐴
2 ssrexf.2 . . . . 5 𝑥𝐵
31, 2dfssf 3986 . . . 4 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
43biimpi 216 . . 3 (𝐴𝐵 → ∀𝑥(𝑥𝐴𝑥𝐵))
5 pm3.45 622 . . . 4 ((𝑥𝐴𝑥𝐵) → ((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
65alimi 1808 . . 3 (∀𝑥(𝑥𝐴𝑥𝐵) → ∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)))
7 moim 2542 . . 3 (∀𝑥((𝑥𝐴𝜑) → (𝑥𝐵𝜑)) → (∃*𝑥(𝑥𝐵𝜑) → ∃*𝑥(𝑥𝐴𝜑)))
84, 6, 73syl 18 . 2 (𝐴𝐵 → (∃*𝑥(𝑥𝐵𝜑) → ∃*𝑥(𝑥𝐴𝜑)))
9 df-rmo 3378 . 2 (∃*𝑥𝐵 𝜑 ↔ ∃*𝑥(𝑥𝐵𝜑))
10 df-rmo 3378 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
118, 9, 103imtr4g 296 1 (𝐴𝐵 → (∃*𝑥𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535  wcel 2106  ∃*wmo 2536  wnfc 2888  ∃*wrmo 3377  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-mo 2538  df-clel 2814  df-nfc 2890  df-rmo 3378  df-ss 3980
This theorem is referenced by:  2sqreunnlem1  27508  2sqreunnlem2  27514  disjss1f  32592  upeu  48817
  Copyright terms: Public domain W3C validator