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Theorem morex 2745
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 2327 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 exancom 1513 . . . 4 (∃𝑥(𝑥𝐴𝜑) ↔ ∃𝑥(𝜑𝑥𝐴))
31, 2bitri 177 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝜑𝑥𝐴))
4 nfmo1 1926 . . . . . 6 𝑥∃*𝑥𝜑
5 nfe1 1399 . . . . . 6 𝑥𝑥(𝜑𝑥𝐴)
64, 5nfan 1471 . . . . 5 𝑥(∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴))
7 mopick 1992 . . . . 5 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → (𝜑𝑥𝐴))
86, 7alrimi 1429 . . . 4 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → ∀𝑥(𝜑𝑥𝐴))
9 morex.1 . . . . 5 𝐵 ∈ V
10 morex.2 . . . . . 6 (𝑥 = 𝐵 → (𝜑𝜓))
11 eleq1 2114 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
1210, 11imbi12d 227 . . . . 5 (𝑥 = 𝐵 → ((𝜑𝑥𝐴) ↔ (𝜓𝐵𝐴)))
139, 12spcv 2661 . . . 4 (∀𝑥(𝜑𝑥𝐴) → (𝜓𝐵𝐴))
148, 13syl 14 . . 3 ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑𝑥𝐴)) → (𝜓𝐵𝐴))
153, 14sylan2b 275 . 2 ((∃*𝑥𝜑 ∧ ∃𝑥𝐴 𝜑) → (𝜓𝐵𝐴))
1615ancoms 259 1 ((∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝜑) → (𝜓𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1255   = wceq 1257  wex 1395  wcel 1407  ∃*wmo 1915  wrex 2322  Vcvv 2572
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-rex 2327  df-v 2574
This theorem is referenced by: (None)
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