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Theorem ssralv2 43594
Description: Quantification restricted to a subclass for two quantifiers. ssralv 4050 for two quantifiers. The proof of ssralv2 43594 was automatically generated by minimizing the automatically translated proof of ssralv2VD 43929. 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 1917 . 2 𝑥(𝐴𝐵𝐶𝐷)
2 nfra1 3281 . 2 𝑥𝑥𝐵𝑦𝐷 𝜑
3 ssralv 4050 . . . . . 6 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐷 𝜑))
43adantr 481 . . . . 5 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐷 𝜑))
5 df-ral 3062 . . . . 5 (∀𝑥𝐴𝑦𝐷 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑))
64, 5imbitrdi 250 . . . 4 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑)))
7 sp 2176 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑) → (𝑥𝐴 → ∀𝑦𝐷 𝜑))
86, 7syl6 35 . . 3 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → (𝑥𝐴 → ∀𝑦𝐷 𝜑)))
9 ssralv 4050 . . . 4 (𝐶𝐷 → (∀𝑦𝐷 𝜑 → ∀𝑦𝐶 𝜑))
109adantl 482 . . 3 ((𝐴𝐵𝐶𝐷) → (∀𝑦𝐷 𝜑 → ∀𝑦𝐶 𝜑))
118, 10syl6d 75 . 2 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → (𝑥𝐴 → ∀𝑦𝐶 𝜑)))
121, 2, 11ralrimd 3261 1 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539  wcel 2106  wral 3061  wss 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-v 3476  df-in 3955  df-ss 3965
This theorem is referenced by:  ordelordALT  43600  ordelordALTVD  43930
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