Theorem issubgilem 8121

Description: Lemma for issubgi 8122.

Hypothesis

Ref	Expression
issubgilem.1	|- ((x e. Y /\ y e. Y) -> (xHy) = (xGy))

Assertion

Ref	Expression
issubgilem	|- ((A e. Y /\ B e. Y) -> (AHB) = (AGB))

Distinct variable groups:   x,A,y   y,B   x,G,y   x,H,y   x,Y,y

Proof of Theorem issubgilem

Step	Hyp	Ref	Expression
1		opreq1 3968	. . 3 |- (x = A -> (xHy) = (AHy))
2		opreq1 3968	. . 3 |- (x = A -> (xGy) = (AGy))
3	1, 2	eqeq12d 1489	. 2 |- (x = A -> ((xHy) = (xGy) <-> (AHy) = (AGy)))
4		opreq2 3969	. . 3 |- (y = B -> (AHy) = (AHB))
5		opreq2 3969	. . 3 |- (y = B -> (AGy) = (AGB))
6	4, 5	eqeq12d 1489	. 2 |- (y = B -> ((AHy) = (AGy) <-> (AHB) = (AGB)))
7		issubgilem.1	. 2 |- ((x e. Y /\ y e. Y) -> (xHy) = (xGy))
8	3, 6, 7	vtocl2ga 1853	1 |- ((A e. Y /\ B e. Y) -> (AHB) = (AGB))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  (class class class)co 3963

This theorem is referenced by:  issubgi 8122

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965

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