Higher-Order Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HOLE Home > Th. List > mpd | GIF version |
Description: Modus ponens. |
Ref | Expression |
---|---|
mp.1 | ⊢ T:∗ |
mp.2 | ⊢ R⊧S |
mp.3 | ⊢ R⊧[S ⇒ T] |
Ref | Expression |
---|---|
mpd | ⊢ R⊧T |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mp.2 | . . . 4 ⊢ R⊧S | |
2 | mp.3 | . . . . 5 ⊢ R⊧[S ⇒ T] | |
3 | 1 | ax-cb1 29 | . . . . . 6 ⊢ R:∗ |
4 | 1 | ax-cb2 30 | . . . . . . 7 ⊢ S:∗ |
5 | mp.1 | . . . . . . 7 ⊢ T:∗ | |
6 | 4, 5 | imval 136 | . . . . . 6 ⊢ ⊤⊧[[S ⇒ T] = [[S ∧ T] = S]] |
7 | 3, 6 | a1i 28 | . . . . 5 ⊢ R⊧[[S ⇒ T] = [[S ∧ T] = S]] |
8 | 2, 7 | mpbi 72 | . . . 4 ⊢ R⊧[[S ∧ T] = S] |
9 | 1, 8 | mpbir 77 | . . 3 ⊢ R⊧[S ∧ T] |
10 | 4, 5 | dfan2 144 | . . . 4 ⊢ ⊤⊧[[S ∧ T] = (S, T)] |
11 | 3, 10 | a1i 28 | . . 3 ⊢ R⊧[[S ∧ T] = (S, T)] |
12 | 9, 11 | mpbi 72 | . 2 ⊢ R⊧(S, T) |
13 | 12 | simprd 36 | 1 ⊢ R⊧T |
Colors of variables: type var term |
Syntax hints: ∗hb 3 = ke 7 [kbr 9 kct 10 ⊧wffMMJ2 11 wffMMJ2t 12 ∧ tan 109 ⇒ tim 111 |
This theorem was proved from axioms: ax-syl 15 ax-jca 17 ax-simpl 20 ax-simpr 21 ax-id 24 ax-trud 26 ax-cb1 29 ax-cb2 30 ax-refl 39 ax-eqmp 42 ax-ded 43 ax-ceq 46 ax-beta 60 ax-distrc 61 ax-leq 62 ax-distrl 63 ax-hbl1 93 ax-17 95 ax-inst 103 |
This theorem depends on definitions: df-ov 65 df-an 118 df-im 119 |
This theorem is referenced by: imp 147 notval2 149 notnot1 150 ecase 153 olc 154 orc 155 exlimdv2 156 ax4e 158 exlimd 171 ac 184 exmid 186 ax2 191 axmp 193 ax5 194 ax11 201 |
Copyright terms: Public domain | W3C validator |