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Theorem r19.9rmv 3307
 Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
Assertion
Ref Expression
r19.9rmv (y y A → (φx A φ))
Distinct variable groups:   x,A   y,A   φ,x
Allowed substitution hint:   φ(y)

Proof of Theorem r19.9rmv
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2097 . . 3 (𝑎 = y → (𝑎 Ay A))
21cbvexv 1792 . 2 (𝑎 𝑎 Ay y A)
3 eleq1 2097 . . . 4 (𝑎 = x → (𝑎 Ax A))
43cbvexv 1792 . . 3 (𝑎 𝑎 Ax x A)
5 df-rex 2306 . . . . 5 (x A φx(x A φ))
6 19.41v 1779 . . . . 5 (x(x A φ) ↔ (x x A φ))
75, 6bitri 173 . . . 4 (x A φ ↔ (x x A φ))
87baibr 828 . . 3 (x x A → (φx A φ))
94, 8sylbi 114 . 2 (𝑎 𝑎 A → (φx A φ))
102, 9sylbir 125 1 (y y A → (φx A φ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301 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-cleq 2030  df-clel 2033  df-rex 2306 This theorem is referenced by:  r19.45mv  3309  iunconstm  3656  fconstfvm  5322  ltexprlemloc  6580
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