Description: Introducing any antecedent in a restricted universal quantification. (Contributed by Glauco Siliprandi, 3-Mar-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | ralimralim | |- ( A. x e. A ph -> A. x e. A ( ps -> ph ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra1 | |- F/ x A. x e. A ph |
|
2 | rspa | |- ( ( A. x e. A ph /\ x e. A ) -> ph ) |
|
3 | ax-1 | |- ( ph -> ( ps -> ph ) ) |
|
4 | 2 3 | syl | |- ( ( A. x e. A ph /\ x e. A ) -> ( ps -> ph ) ) |
5 | 4 | ex | |- ( A. x e. A ph -> ( x e. A -> ( ps -> ph ) ) ) |
6 | 1 5 | ralrimi | |- ( A. x e. A ph -> A. x e. A ( ps -> ph ) ) |