Description: Rule used to change the bound variables and classes in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by RP, 17-Jul-2020)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cbviuneq12df.xph | |- F/ x ph |
|
| cbviuneq12df.yph | |- F/ y ph |
||
| cbviuneq12df.x | |- F/_ x X |
||
| cbviuneq12df.y | |- F/_ y Y |
||
| cbviuneq12df.xa | |- F/_ x A |
||
| cbviuneq12df.ya | |- F/_ y A |
||
| cbviuneq12df.b | |- F/_ y B |
||
| cbviuneq12df.xc | |- F/_ x C |
||
| cbviuneq12df.yc | |- F/_ y C |
||
| cbviuneq12df.d | |- F/_ x D |
||
| cbviuneq12df.f | |- F/_ x F |
||
| cbviuneq12df.g | |- F/_ y G |
||
| cbviuneq12df.xel | |- ( ( ph /\ y e. C ) -> X e. A ) |
||
| cbviuneq12df.yel | |- ( ( ph /\ x e. A ) -> Y e. C ) |
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| cbviuneq12df.xsub | |- ( ( ph /\ y e. C /\ x = X ) -> B = F ) |
||
| cbviuneq12df.ysub | |- ( ( ph /\ x e. A /\ y = Y ) -> D = G ) |
||
| cbviuneq12df.eq1 | |- ( ( ph /\ x e. A ) -> B = G ) |
||
| cbviuneq12df.eq2 | |- ( ( ph /\ y e. C ) -> D = F ) |
||
| Assertion | cbviuneq12df | |- ( ph -> U_ x e. A B = U_ y e. C D ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbviuneq12df.xph | |- F/ x ph |
|
| 2 | cbviuneq12df.yph | |- F/ y ph |
|
| 3 | cbviuneq12df.x | |- F/_ x X |
|
| 4 | cbviuneq12df.y | |- F/_ y Y |
|
| 5 | cbviuneq12df.xa | |- F/_ x A |
|
| 6 | cbviuneq12df.ya | |- F/_ y A |
|
| 7 | cbviuneq12df.b | |- F/_ y B |
|
| 8 | cbviuneq12df.xc | |- F/_ x C |
|
| 9 | cbviuneq12df.yc | |- F/_ y C |
|
| 10 | cbviuneq12df.d | |- F/_ x D |
|
| 11 | cbviuneq12df.f | |- F/_ x F |
|
| 12 | cbviuneq12df.g | |- F/_ y G |
|
| 13 | cbviuneq12df.xel | |- ( ( ph /\ y e. C ) -> X e. A ) |
|
| 14 | cbviuneq12df.yel | |- ( ( ph /\ x e. A ) -> Y e. C ) |
|
| 15 | cbviuneq12df.xsub | |- ( ( ph /\ y e. C /\ x = X ) -> B = F ) |
|
| 16 | cbviuneq12df.ysub | |- ( ( ph /\ x e. A /\ y = Y ) -> D = G ) |
|
| 17 | cbviuneq12df.eq1 | |- ( ( ph /\ x e. A ) -> B = G ) |
|
| 18 | cbviuneq12df.eq2 | |- ( ( ph /\ y e. C ) -> D = F ) |
|
| 19 | eqimss | |- ( B = G -> B C_ G ) |
|
| 20 | 17 19 | syl | |- ( ( ph /\ x e. A ) -> B C_ G ) |
| 21 | 1 2 4 6 7 8 9 10 12 14 16 20 | ss2iundf | |- ( ph -> U_ x e. A B C_ U_ y e. C D ) |
| 22 | eqimss | |- ( D = F -> D C_ F ) |
|
| 23 | 18 22 | syl | |- ( ( ph /\ y e. C ) -> D C_ F ) |
| 24 | 2 1 3 8 10 6 5 7 11 13 15 23 | ss2iundf | |- ( ph -> U_ y e. C D C_ U_ x e. A B ) |
| 25 | 21 24 | eqssd | |- ( ph -> U_ x e. A B = U_ y e. C D ) |