Description: A specialized lemma for set theory (to derive the Axiom of Pairing). (Contributed by NM, 18-Oct-1995) (Proof shortened by Andrew Salmon, 13-May-2011) (Proof shortened by Wolf Lammen, 5-Jan-2013)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | prlem1.1 | |- ( ph -> ( et <-> ch ) ) |
|
| prlem1.2 | |- ( ps -> -. th ) |
||
| Assertion | prlem1 | |- ( ph -> ( ps -> ( ( ( ps /\ ch ) \/ ( th /\ ta ) ) -> et ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | prlem1.1 | |- ( ph -> ( et <-> ch ) ) |
|
| 2 | prlem1.2 | |- ( ps -> -. th ) |
|
| 3 | 1 | biimprd | |- ( ph -> ( ch -> et ) ) |
| 4 | 3 | adantld | |- ( ph -> ( ( ps /\ ch ) -> et ) ) |
| 5 | 2 | pm2.21d | |- ( ps -> ( th -> et ) ) |
| 6 | 5 | adantrd | |- ( ps -> ( ( th /\ ta ) -> et ) ) |
| 7 | 4 6 | jaao | |- ( ( ph /\ ps ) -> ( ( ( ps /\ ch ) \/ ( th /\ ta ) ) -> et ) ) |
| 8 | 7 | ex | |- ( ph -> ( ps -> ( ( ( ps /\ ch ) \/ ( th /\ ta ) ) -> et ) ) ) |