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Theorem spc2gv 2637
 Description: Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)
Hypothesis
Ref Expression
spc2egv.1 ((x = A y = B) → (φψ))
Assertion
Ref Expression
spc2gv ((A 𝑉 B 𝑊) → (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 2562 . . . 4 (A 𝑉x x = A)
2 elisset 2562 . . . 4 (B 𝑊y y = B)
31, 2anim12i 321 . . 3 ((A 𝑉 B 𝑊) → (x x = A y y = B))
4 eeanv 1804 . . 3 (xy(x = A y = B) ↔ (x x = A y y = B))
53, 4sylibr 137 . 2 ((A 𝑉 B 𝑊) → xy(x = A y = B))
6 spc2egv.1 . . . . . 6 ((x = A y = B) → (φψ))
76biimpcd 148 . . . . 5 (φ → ((x = A y = B) → ψ))
872alimi 1342 . . . 4 (xyφxy((x = A y = B) → ψ))
9 exim 1487 . . . . 5 (y((x = A y = B) → ψ) → (y(x = A y = B) → yψ))
109alimi 1341 . . . 4 (xy((x = A y = B) → ψ) → x(y(x = A y = B) → yψ))
11 exim 1487 . . . 4 (x(y(x = A y = B) → yψ) → (xy(x = A y = B) → xyψ))
128, 10, 113syl 17 . . 3 (xyφ → (xy(x = A y = B) → xyψ))
13 19.9v 1748 . . . 4 (xyψyψ)
14 19.9v 1748 . . . 4 (yψψ)
1513, 14bitri 173 . . 3 (xyψψ)
1612, 15syl6ib 150 . 2 (xyφ → (xy(x = A y = B) → ψ))
175, 16syl5com 26 1 ((A 𝑉 B 𝑊) → (xyφψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553 This theorem is referenced by:  trel  3852  elovmpt2  5643
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