Description: A more efficient proof of Theorem 9.20 of Clemente p. 45. Compare with ex-natded9.20 . (Contributed by David A. Wheeler, 19-Feb-2017) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | ex-natded9.20.1 | |- ( ph -> ( ps /\ ( ch \/ th ) ) ) |
|
| Assertion | ex-natded9.20-2 | |- ( ph -> ( ( ps /\ ch ) \/ ( ps /\ th ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ex-natded9.20.1 | |- ( ph -> ( ps /\ ( ch \/ th ) ) ) |
|
| 2 | 1 | simpld | |- ( ph -> ps ) |
| 3 | 2 | anim1i | |- ( ( ph /\ ch ) -> ( ps /\ ch ) ) |
| 4 | 3 | orcd | |- ( ( ph /\ ch ) -> ( ( ps /\ ch ) \/ ( ps /\ th ) ) ) |
| 5 | 2 | anim1i | |- ( ( ph /\ th ) -> ( ps /\ th ) ) |
| 6 | 5 | olcd | |- ( ( ph /\ th ) -> ( ( ps /\ ch ) \/ ( ps /\ th ) ) ) |
| 7 | 1 | simprd | |- ( ph -> ( ch \/ th ) ) |
| 8 | 4 6 7 | mpjaodan | |- ( ph -> ( ( ps /\ ch ) \/ ( ps /\ th ) ) ) |