Theorem issubgilem 8117

Description: Lemma for issubgi 8118.

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 3974	. . 3 |- (x = A -> (xHy) = (AHy))
2		opreq1 3974	. . 3 |- (x = A -> (xGy) = (AGy))
3	1, 2	eqeq12d 1492	. 2 |- (x = A -> ((xHy) = (xGy) <-> (AHy) = (AGy)))
4		opreq2 3975	. . 3 |- (y = B -> (AHy) = (AHB))
5		opreq2 3975	. . 3 |- (y = B -> (AGy) = (AGB))
6	4, 5	eqeq12d 1492	. 2 |- (y = B -> ((AHy) = (AGy) <-> (AHB) = (AGB)))
7		issubgilem.1	. 2 |- ((x e. Y /\ y e. Y) -> (xHy) = (xGy))
8	3, 6, 7	vtocl2ga 1856	1 |- ((A e. Y /\ B e. Y) -> (AHB) = (AGB))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  (class class class)co 3969

This theorem is referenced by:  issubgi 8118

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971

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