Theorem ax4g 149

Description: If F is true for all x:al, then it is true for A. (Contributed by Mario Carneiro, 9-Oct-2014.)

Hypotheses

Ref	Expression
ax4g.1	|- F:(al -> *)
ax4g.2	|- A:al

Assertion

Ref	Expression
ax4g	|- (A.F) |= (FA)

Proof of Theorem ax4g

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wal 134	. . . 4 |- A.:((al -> *) -> *)
2		ax4g.1	. . . 4 |- F:(al -> *)
3	1, 2	wc 50	. . 3 |- (A.F):*
4	3	trud 27	. 2 |- (A.F) |= T.
5		ax4g.2	. . . 4 |- A:al
6	2, 5	wc 50	. . 3 |- (FA):*
7	4	ax-cb1 29	. . . . . 6 |- (A.F):*
8	7	id 25	. . . . 5 |- (A.F) |= (A.F)
9	2	alval 142	. . . . . 6 |- T. |= [(A.F) = [F = \p:al T.]]
10	7, 9	a1i 28	. . . . 5 |- (A.F) |= [(A.F) = [F = \p:al T.]]
11	8, 10	mpbi 82	. . . 4 |- (A.F) |= [F = \p:al T.]
12	2, 5, 11	ceq1 89	. . 3 |- (A.F) |= [(FA) = (\p:al T.A)]
13	5, 4	hbth 109	. . 3 |- (A.F) |= [(\p:al T.A) = T.]
14	6, 12, 13	eqtri 95	. 2 |- (A.F) |= [(FA) = T.]
15	4, 14	mpbir 87	1 |- (A.F) |= (FA)

Syntax hints:   -> ht 2  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  A.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

This theorem is referenced by:  ax4  150  cla4v  152