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

Theorem morex 3588
Description: Derive membership from uniqueness. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
morex.1 𝐵 ∈ V
morex.2 (𝑥 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
morex ((∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓𝐵𝐴))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem morex
StepHypRef Expression
1 df-rex 3102 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 exancom 1947 . . . 4 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑥(𝜑𝑥𝐴))
31, 2bitri 266 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝜑𝑥𝐴))
4 nfmo1 2643 . . . . . 6 𝑥∃*𝑥𝜑
5 nfe1 2194 . . . . . 6 𝑥𝑥(𝜑𝑥𝐴)
64, 5nfan 1990 . . . . 5 𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴))
7 mopick 2699 . . . . 5 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → (𝜑𝑥𝐴))
86, 7alrimi 2249 . . . 4 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → ∀𝑥(𝜑𝑥𝐴))
9 morex.1 . . . . 5 𝐵 ∈ V
10 morex.2 . . . . . 6 (𝑥 = 𝐵 → (𝜑𝜓))
11 eleq1 2873 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
1210, 11imbi12d 335 . . . . 5 (𝑥 = 𝐵 → ((𝜑𝑥𝐴) ↔ (𝜓𝐵𝐴)))
139, 12spcv 3492 . . . 4 (∀𝑥(𝜑𝑥𝐴) → (𝜓𝐵𝐴))
148, 13syl 17 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → (𝜓𝐵𝐴))
153, 14sylan2b 583 . 2 ((∃*𝑥𝜑 ∧ ∃𝑥𝐴 𝜑) → (𝜓𝐵𝐴))
1615ancoms 448 1 ((∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wex 1859  wcel 2156  ∃*wmo 2631  wrex 3097  Vcvv 3391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-rex 3102  df-v 3393
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator