Theorem issubgoilem 31623

Description: Lemma for hhssabloilem 31624. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)

Hypothesis

Ref	Expression
issubgoilem.1	⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦))

Assertion

Ref	Expression
issubgoilem	⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (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

Step	Hyp	Ref	Expression
1		oveq1 7419	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
2		oveq1 7419	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
3	1, 2	eqeq12d 2778	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑥𝐺𝑦) ↔ (𝐴𝐻𝑦) = (𝐴𝐺𝑦)))
4		oveq2 7420	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
5		oveq2 7420	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
6	4, 5	eqeq12d 2778	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝐴𝐺𝑦) ↔ (𝐴𝐻𝐵) = (𝐴𝐺𝐵)))
7		issubgoilem.1	. 2 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦))
8	3, 6, 7	vtocl2ga 3541	1 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  (class class class)co 7412

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7415

This theorem is used by: (None)