Theorem issubgoilem 30982

Description: Lemma for hhssabloilem 30983. (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 7408	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
2		oveq1 7408	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦))
3	1, 2	eqeq12d 2740	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑦) = (𝑥𝐺𝑦) ↔ (𝐴𝐻𝑦) = (𝐴𝐺𝑦)))
4		oveq2 7409	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
5		oveq2 7409	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵))
6	4, 5	eqeq12d 2740	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴𝐻𝑦) = (𝐴𝐺𝑦) ↔ (𝐴𝐻𝐵) = (𝐴𝐺𝐵)))
7		issubgoilem.1	. 2 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐻𝑦) = (𝑥𝐺𝑦))
8	3, 6, 7	vtocl2ga 3559	1 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝐴𝐻𝐵) = (𝐴𝐺𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  (class class class)co 7401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404

This theorem is referenced by: (None)