Description: Implicit substitution of a class for a setvar variable. (Closed theorem version of vtoclef .) (Contributed by NM, 7-Nov-2005) (Revised by Mario Carneiro, 11-Oct-2016)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | vtoclegft | |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ph ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elisset | |- ( A e. B -> E. x x = A ) |
|
| 2 | exim | |- ( A. x ( x = A -> ph ) -> ( E. x x = A -> E. x ph ) ) |
|
| 3 | 1 2 | mpan9 | |- ( ( A e. B /\ A. x ( x = A -> ph ) ) -> E. x ph ) |
| 4 | 3 | 3adant2 | |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> E. x ph ) |
| 5 | 19.9t | |- ( F/ x ph -> ( E. x ph <-> ph ) ) |
|
| 6 | 5 | 3ad2ant2 | |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ( E. x ph <-> ph ) ) |
| 7 | 4 6 | mpbid | |- ( ( A e. B /\ F/ x ph /\ A. x ( x = A -> ph ) ) -> ph ) |