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Theorem ssralv2 43340
Description: Quantification restricted to a subclass for two quantifiers. ssralv 4051 for two quantifiers. The proof of ssralv2 43340 was automatically generated by minimizing the automatically translated proof of ssralv2VD 43675. The automatic translation is by the tools program translate_without_overwriting.cmd. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ssralv2 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑦,𝐶   𝑥,𝐷   𝑦,𝐷
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem ssralv2
StepHypRef Expression
1 nfv 1918 . 2 𝑥(𝐴𝐵𝐶𝐷)
2 nfra1 3282 . 2 𝑥𝑥𝐵𝑦𝐷 𝜑
3 ssralv 4051 . . . . . 6 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐷 𝜑))
43adantr 482 . . . . 5 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐷 𝜑))
5 df-ral 3063 . . . . 5 (∀𝑥𝐴𝑦𝐷 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑))
64, 5imbitrdi 250 . . . 4 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑)))
7 sp 2177 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑) → (𝑥𝐴 → ∀𝑦𝐷 𝜑))
86, 7syl6 35 . . 3 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → (𝑥𝐴 → ∀𝑦𝐷 𝜑)))
9 ssralv 4051 . . . 4 (𝐶𝐷 → (∀𝑦𝐷 𝜑 → ∀𝑦𝐶 𝜑))
109adantl 483 . . 3 ((𝐴𝐵𝐶𝐷) → (∀𝑦𝐷 𝜑 → ∀𝑦𝐶 𝜑))
118, 10syl6d 75 . 2 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → (𝑥𝐴 → ∀𝑦𝐶 𝜑)))
121, 2, 11ralrimd 3262 1 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wcel 2107  wral 3062  wss 3949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-in 3956  df-ss 3966
This theorem is referenced by:  ordelordALT  43346  ordelordALTVD  43676
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