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Theorem r19.28m 3288
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
Hypothesis
Ref Expression
r19.3rmOLD.1 xφ
Assertion
Ref Expression
r19.28m (x x A → (x A (φ ψ) ↔ (φ x A ψ)))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem r19.28m
StepHypRef Expression
1 r19.3rmOLD.1 . . . 4 xφ
21r19.3rmOLD 3287 . . 3 (x x A → (φx A φ))
32anbi1d 441 . 2 (x x A → ((φ x A ψ) ↔ (x A φ x A ψ)))
4 r19.26 2419 . 2 (x A (φ ψ) ↔ (x A φ x A ψ))
53, 4syl6rbbr 188 1 (x x A → (x A (φ ψ) ↔ (φ x A ψ)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wnf 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-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-4 1381  ax-ial 1409  ax-i5r 1410
This theorem depends on definitions:  df-bi 110  df-nf 1330  df-ral 2289
This theorem is referenced by:  r19.28mv  3293  raaanlem  3305
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