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Theorem rr19.28v 2677
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 29-Oct-2012.)
Assertion
Ref Expression
rr19.28v (x A y A (φ ψ) ↔ x A (φ y A ψ))
Distinct variable groups:   y,A   x,y   φ,y
Allowed substitution hints:   φ(x)   ψ(x,y)   A(x)

Proof of Theorem rr19.28v
StepHypRef Expression
1 simpl 102 . . . . . 6 ((φ ψ) → φ)
21ralimi 2378 . . . . 5 (y A (φ ψ) → y A φ)
3 biidd 161 . . . . . 6 (y = x → (φφ))
43rspcv 2646 . . . . 5 (x A → (y A φφ))
52, 4syl5 28 . . . 4 (x A → (y A (φ ψ) → φ))
6 simpr 103 . . . . . 6 ((φ ψ) → ψ)
76ralimi 2378 . . . . 5 (y A (φ ψ) → y A ψ)
87a1i 9 . . . 4 (x A → (y A (φ ψ) → y A ψ))
95, 8jcad 291 . . 3 (x A → (y A (φ ψ) → (φ y A ψ)))
109ralimia 2376 . 2 (x A y A (φ ψ) → x A (φ y A ψ))
11 r19.28av 2443 . . 3 ((φ y A ψ) → y A (φ ψ))
1211ralimi 2378 . 2 (x A (φ y A ψ) → x A y A (φ ψ))
1310, 12impbii 117 1 (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   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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  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-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553
This theorem is referenced by: (None)
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