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.

Step	Hyp	Ref	Expression
1		wfal 135	. . . 4 ⊢ ⊥:∗
2	1	id 25	. . 3 ⊢ ⊥⊧⊥
3		df-fal 127	. . . 4 ⊢ ⊤⊧[⊥ = (∀λp:∗ p:∗)]
4	1, 3	a1i 28	. . 3 ⊢ ⊥⊧[⊥ = (∀λp:∗ p:∗)]
5	2, 4	mpbi 82	. 2 ⊢ ⊥⊧(∀λp:∗ p:∗)
6		wv 64	. . 3 ⊢ p:∗:∗
7		pm2.21.1	. . 3 ⊢ A:∗
8	6, 7	weqi 76	. . . 4 ⊢ [p:∗ = A]:∗
9	8	id 25	. . 3 ⊢ [p:∗ = A]⊧[p:∗ = A]
10	6, 7, 9	cla4v 152	. 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 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