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Theorem pm2.21 153
 Description: A falsehood implies anything. (Contributed by Mario Carneiro, 9-Oct-2014.)
Hypothesis
Ref Expression
pm2.21.1 A:∗
Assertion
Ref Expression
pm2.21 ⊥⊧A

Proof of Theorem pm2.21
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 wfal 135 . . . 4 ⊥:∗
21id 25 . . 3 ⊥⊧⊥
3 df-fal 127 . . . 4 ⊤⊧[⊥ = (λp:∗ p:∗)]
41, 3a1i 28 . . 3 ⊥⊧[⊥ = (λp:∗ p:∗)]
52, 4mpbi 82 . 2 ⊥⊧(λp:∗ p:∗)
6 wv 64 . . 3 p:∗:∗
7 pm2.21.1 . . 3 A:∗
86, 7weqi 76 . . . 4 [p:∗ = A]:∗
98id 25 . . 3 [p:∗ = A]⊧[p:∗ = A]
106, 7, 9cla4v 152 . 2 (λp:∗ p:∗)⊧A
115, 10syl 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 118  ∀tal 122 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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113 This theorem depends on definitions:  df-ov 73  df-al 126  df-fal 127 This theorem is referenced by:  notval2  159  notnot  200  ax3  205
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