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Theorem cgsex2g 2584
Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Jul-1995.)
Hypotheses
Ref Expression
cgsex2g.1 ((x = A y = B) → χ)
cgsex2g.2 (χ → (φψ))
Assertion
Ref Expression
cgsex2g ((A 𝑉 B 𝑊) → (xy(χ φ) ↔ ψ))
Distinct variable groups:   x,y,ψ   x,A,y   x,B,y
Allowed substitution hints:   φ(x,y)   χ(x,y)   𝑉(x,y)   𝑊(x,y)

Proof of Theorem cgsex2g
StepHypRef Expression
1 cgsex2g.2 . . . 4 (χ → (φψ))
21biimpa 280 . . 3 ((χ φ) → ψ)
32exlimivv 1773 . 2 (xy(χ φ) → ψ)
4 elisset 2562 . . . . . 6 (A 𝑉x x = A)
5 elisset 2562 . . . . . 6 (B 𝑊y y = B)
64, 5anim12i 321 . . . . 5 ((A 𝑉 B 𝑊) → (x x = A y y = B))
7 eeanv 1804 . . . . 5 (xy(x = A y = B) ↔ (x x = A y y = B))
86, 7sylibr 137 . . . 4 ((A 𝑉 B 𝑊) → xy(x = A y = B))
9 cgsex2g.1 . . . . 5 ((x = A y = B) → χ)
1092eximi 1489 . . . 4 (xy(x = A y = B) → xyχ)
118, 10syl 14 . . 3 ((A 𝑉 B 𝑊) → xyχ)
121biimprcd 149 . . . . 5 (ψ → (χφ))
1312ancld 308 . . . 4 (ψ → (χ → (χ φ)))
14132eximdv 1759 . . 3 (ψ → (xyχxy(χ φ)))
1511, 14syl5com 26 . 2 ((A 𝑉 B 𝑊) → (ψxy(χ φ)))
163, 15impbid2 131 1 ((A 𝑉 B 𝑊) → (xy(χ φ) ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = 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: (None)
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