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Theorem issubgoilem 29608
Description: Lemma for hhssabloilem 29609. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
issubgoilem.1 ((𝑥𝑌𝑦𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦))
Assertion
Ref Expression
issubgoilem ((𝐴𝑌𝐵𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝑌,𝑦
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem issubgoilem
StepHypRef Expression
1 oveq1 7275 . . 3 (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
2 oveq1 7275 . . 3 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
31, 2eqeq12d 2754 . 2 (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑥𝐺𝑦) ↔ (𝐴𝐻𝑦) = (𝐴𝐺𝑦)))
4 oveq2 7276 . . 3 (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
5 oveq2 7276 . . 3 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
64, 5eqeq12d 2754 . 2 (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝐴𝐺𝑦) ↔ (𝐴𝐻𝐵) = (𝐴𝐺𝐵)))
7 issubgoilem.1 . 2 ((𝑥𝑌𝑦𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦))
83, 6, 7vtocl2ga 3512 1 ((𝐴𝑌𝐵𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  (class class class)co 7268
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-iota 6385  df-fv 6435  df-ov 7271
This theorem is referenced by: (None)
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