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Theorem rr19.3v 2980
 Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. We don't need the non-empty class condition of r19.3rzv 3643 when there is an outer quantifier. (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 228 . . . 4 (y = x → (φφ))
21rspcv 2951 . . 3 (x A → (y A φφ))
32ralimia 2687 . 2 (x A y A φx A φ)
4 ax-1 5 . . . 4 (φ → (y Aφ))
54ralrimiv 2696 . . 3 (φy A φ)
65ralimi 2689 . 2 (x A φx A y A φ)
73, 6impbii 180 1 (x A y A φx A φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2614 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861 This theorem is referenced by: (None)
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