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Theorem r19.3rm 3288
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
Hypothesis
Ref Expression
r19.3rm.1 xφ
Assertion
Ref Expression
r19.3rm (y y A → (φx A φ))
Distinct variable groups:   x,A   y,A
Allowed substitution hints:   φ(x,y)

Proof of Theorem r19.3rm
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2083 . . 3 (𝑎 = y → (𝑎 Ay A))
21cbvexv 1778 . 2 (𝑎 𝑎 Ay y A)
3 eleq1 2083 . . . 4 (𝑎 = x → (𝑎 Ax A))
43cbvexv 1778 . . 3 (𝑎 𝑎 Ax x A)
5 biimt 230 . . . 4 (x x A → (φ ↔ (x x Aφ)))
6 df-ral 2288 . . . . 5 (x A φx(x Aφ))
7 r19.3rm.1 . . . . . 6 xφ
8719.23 1551 . . . . 5 (x(x Aφ) ↔ (x x Aφ))
96, 8bitri 173 . . . 4 (x A φ ↔ (x x Aφ))
105, 9syl6bbr 187 . . 3 (x x A → (φx A φ))
114, 10sylbi 114 . 2 (𝑎 𝑎 A → (φx A φ))
122, 11sylbir 125 1 (y y A → (φx A φ))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1226  wnf 1329  wex 1363   wcel 1375  wral 2283
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 1364  ax-ie2 1365  ax-8 1377  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-cleq 2016  df-clel 2019  df-ral 2288
This theorem is referenced by:  r19.3rmv  3289
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