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Theorem raaanlem 3320
Description: Special case of raaan 3321 where A is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
Hypotheses
Ref Expression
raaan.1 yφ
raaan.2 xψ
Assertion
Ref Expression
raaanlem (x x A → (x A y A (φ ψ) ↔ (x A φ y A ψ)))
Distinct variable group:   x,y,A
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem raaanlem
StepHypRef Expression
1 eleq1 2097 . . . 4 (x = y → (x Ay A))
21cbvexv 1792 . . 3 (x x Ay y A)
3 raaan.1 . . . . 5 yφ
43r19.28m 3305 . . . 4 (y y A → (y A (φ ψ) ↔ (φ y A ψ)))
54ralbidv 2320 . . 3 (y y A → (x A y A (φ ψ) ↔ x A (φ y A ψ)))
62, 5sylbi 114 . 2 (x x A → (x A y A (φ ψ) ↔ x A (φ y A ψ)))
7 nfcv 2175 . . . 4 xA
8 raaan.2 . . . 4 xψ
97, 8nfralxy 2354 . . 3 xy A ψ
109r19.27m 3310 . 2 (x x A → (x A (φ y A ψ) ↔ (x A φ y A ψ)))
116, 10bitrd 177 1 (x x A → (x A y A (φ ψ) ↔ (x A φ y A ψ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wnf 1346  wex 1378   wcel 1390  wral 2300
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-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305
This theorem is referenced by:  raaan  3321
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