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Theorem ssralv2 44556
Description: Quantification restricted to a subclass for two quantifiers. ssralv 4051 for two quantifiers. The proof of ssralv2 44556 was automatically generated by minimizing the automatically translated proof of ssralv2VD 44891. 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 1913 . 2 𝑥(𝐴𝐵𝐶𝐷)
2 nfra1 3283 . 2 𝑥𝑥𝐵𝑦𝐷 𝜑
3 ssralv 4051 . . . . . 6 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐷 𝜑))
43adantr 480 . . . . 5 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐷 𝜑))
5 df-ral 3061 . . . . 5 (∀𝑥𝐴𝑦𝐷 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑))
64, 5imbitrdi 251 . . . 4 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥(𝑥𝐴 → ∀𝑦𝐷 𝜑)))
7 sp 2182 . . . 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 481 . . 3 ((𝐴𝐵𝐶𝐷) → (∀𝑦𝐷 𝜑 → ∀𝑦𝐶 𝜑))
118, 10syl6d 75 . 2 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → (𝑥𝐴 → ∀𝑦𝐶 𝜑)))
121, 2, 11ralrimd 3263 1 ((𝐴𝐵𝐶𝐷) → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑥𝐴𝑦𝐶 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1537  wcel 2107  wral 3060  wss 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-12 2176
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1779  df-nf 1783  df-ral 3061  df-ss 3967
This theorem is referenced by:  ordelordALT  44562  ordelordALTVD  44892
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