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Theorem rr19.28v 2981
 Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. We don't need the non-empty class condition of r19.28zv 3645 when there is an outer quantifier. (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 443 . . . . . 6 ((φ ψ) → φ)
21ralimi 2689 . . . . 5 (y A (φ ψ) → y A φ)
3 biidd 228 . . . . . 6 (y = x → (φφ))
43rspcv 2951 . . . . 5 (x A → (y A φφ))
52, 4syl5 28 . . . 4 (x A → (y A (φ ψ) → φ))
6 simpr 447 . . . . . 6 ((φ ψ) → ψ)
76ralimi 2689 . . . . 5 (y A (φ ψ) → y A ψ)
87a1i 10 . . . 4 (x A → (y A (φ ψ) → y A ψ))
95, 8jcad 519 . . 3 (x A → (y A (φ ψ) → (φ y A ψ)))
109ralimia 2687 . 2 (x A y A (φ ψ) → x A (φ y A ψ))
11 r19.28av 2753 . . 3 ((φ y A ψ) → y A (φ ψ))
1211ralimi 2689 . 2 (x A (φ y A ψ) → x A y A (φ ψ))
1310, 12impbii 180 1 (x A y A (φ ψ) ↔ x A (φ y A ψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = 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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