Theorem pm2.21 143

Description: A falsehood implies anything.

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.

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

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