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Theorem rr19.3v 2676
Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.)
Assertion
Ref Expression
rr19.3v (x A y A φx A φ)
Distinct variable groups:   y,A   x,y   φ,y
Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem rr19.3v
StepHypRef Expression
1 biidd 161 . . . 4 (y = x → (φφ))
21rspcv 2646 . . 3 (x A → (y A φφ))
32ralimia 2376 . 2 (x A y A φx A φ)
4 ax-1 5 . . . 4 (φ → (y Aφ))
54ralrimiv 2385 . . 3 (φy A φ)
65ralimi 2378 . 2 (x A φx A y A φ)
73, 6impbii 117 1 (x A y A φx A φ)
Colors of variables: wff set class
Syntax hints:  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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