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Theorem raaanlem 3305
 Description: Special case of raaan 3306 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 2082 . . . 4 (x = y → (x Ay A))
21cbvexv 1777 . . 3 (x x Ay y A)
3 raaan.1 . . . . 5 yφ
43r19.28m 3288 . . . 4 (y y A → (y A (φ ψ) ↔ (φ y A ψ)))
54ralbidv 2304 . . 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 2160 . . . 4 xA
8 raaan.2 . . . 4 xψ
97, 8nfralxy 2338 . . 3 xy A ψ
109r19.27m 3295 . 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  Ⅎwnf 1329  ∃wex 1362   ∈ wcel 1374  ∀wral 2284 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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289 This theorem is referenced by:  raaan  3306
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