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Mirrors > Home > HOLE Home > Th. List > pm2.21 | GIF version |
Description: A falsehood implies anything. |
Ref | Expression |
---|---|
pm2.21.1 | ⊢ A:∗ |
Ref | Expression |
---|---|
pm2.21 | ⊢ ⊥⊧A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfal 125 | . . . 4 ⊢ ⊥:∗ | |
2 | 1 | id 25 | . . 3 ⊢ ⊥⊧⊥ |
3 | df-fal 117 | . . . 4 ⊢ ⊤⊧[⊥ = (∀λp:∗ p:∗)] | |
4 | 1, 3 | a1i 28 | . . 3 ⊢ ⊥⊧[⊥ = (∀λp:∗ p:∗)] |
5 | 2, 4 | mpbi 72 | . 2 ⊢ ⊥⊧(∀λp:∗ p:∗) |
6 | wv 58 | . . 3 ⊢ p:∗:∗ | |
7 | pm2.21.1 | . . 3 ⊢ A:∗ | |
8 | 6, 7 | weqi 68 | . . . 4 ⊢ [p:∗ = A]:∗ |
9 | 8 | id 25 | . . 3 ⊢ [p:∗ = A]⊧[p:∗ = A] |
10 | 6, 7, 9 | cla4v 142 | . 2 ⊢ (∀λp:∗ p:∗)⊧A |
11 | 5, 10 | syl 16 | 1 ⊢ ⊥⊧A |
Colors of variables: type var term |
Syntax hints: tv 1 ∗hb 3 kc 5 λkl 6 = ke 7 [kbr 9 ⊧wffMMJ2 11 wffMMJ2t 12 ⊥tfal 108 ∀tal 112 |
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-hbl1 93 ax-17 95 ax-inst 103 |
This theorem depends on definitions: df-ov 65 df-al 116 df-fal 117 |
This theorem is referenced by: notval2 149 notnot 187 ax3 192 |
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