Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  issubgoilem Structured version   Visualization version   GIF version

Theorem issubgoilem 27501
 Description: Lemma for hhssabloilem 27502. (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 6556 . . 3 (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
2 oveq1 6556 . . 3 (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
31, 2eqeq12d 2625 . 2 (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑥𝐺𝑦) ↔ (𝐴𝐻𝑦) = (𝐴𝐺𝑦)))
4 oveq2 6557 . . 3 (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
5 oveq2 6557 . . 3 (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
64, 5eqeq12d 2625 . 2 (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝐴𝐺𝑦) ↔ (𝐴𝐻𝐵) = (𝐴𝐺𝐵)))
7 issubgoilem.1 . 2 ((𝑥𝑌𝑦𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦))
83, 6, 7vtocl2ga 3247 1 ((𝐴𝑌𝐵𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  (class class class)co 6549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator