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Theorem spc2gv 2776
Description: Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
Hypothesis
Ref Expression
spc2egv.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
spc2gv ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦𝜑𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem spc2gv
StepHypRef Expression
1 elisset 2700 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 2700 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
31, 2anim12i 336 . . 3 ((𝐴𝑉𝐵𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
4 eeanv 1904 . . 3 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
53, 4sylibr 133 . 2 ((𝐴𝑉𝐵𝑊) → ∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵))
6 spc2egv.1 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
76biimpcd 158 . . . . 5 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → 𝜓))
872alimi 1432 . . . 4 (∀𝑥𝑦𝜑 → ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝜓))
9 exim 1578 . . . . 5 (∀𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝜓) → (∃𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑦𝜓))
109alimi 1431 . . . 4 (∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → 𝜓) → ∀𝑥(∃𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑦𝜓))
11 exim 1578 . . . 4 (∀𝑥(∃𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑦𝜓) → (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦𝜓))
128, 10, 113syl 17 . . 3 (∀𝑥𝑦𝜑 → (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦𝜓))
13 19.9v 1843 . . . 4 (∃𝑥𝑦𝜓 ↔ ∃𝑦𝜓)
14 19.9v 1843 . . . 4 (∃𝑦𝜓𝜓)
1513, 14bitri 183 . . 3 (∃𝑥𝑦𝜓𝜓)
1612, 15syl6ib 160 . 2 (∀𝑥𝑦𝜑 → (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → 𝜓))
175, 16syl5com 29 1 ((𝐴𝑉𝐵𝑊) → (∀𝑥𝑦𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1329   = wceq 1331  wex 1468  wcel 1480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-v 2688
This theorem is referenced by:  rspc2gv  2801  trel  4033  exmidundif  4129  exmidundifim  4130  elovmpo  5971  cnmpt12  12466  cnmpt22  12473  exmidsbthrlem  13247
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