Theorem ax4 150

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

Hypothesis

Ref	Expression
ax4.1	|- A:*

Assertion

Ref	Expression
ax4	|- (A.\x:al A) |= A

Proof of Theorem ax4

Step	Hyp	Ref	Expression
1		ax4.1	. . . 4 |- A:*
2	1	wl 66	. . 3 |- \x:al A:(al -> *)
3		wv 64	. . 3 |- x:al:al
4	2, 3	ax4g 149	. 2 |- (A.\x:al A) |= (\x:al Ax:al)
5	4	ax-cb1 29	. . 3 |- (A.\x:al A):*
6	1	beta 92	. . 3 |- T. |= [(\x:al Ax:al) = A]
7	5, 6	a1i 28	. 2 |- (A.\x:al A) |= [(\x:al Ax:al) = A]
8	4, 7	mpbi 82	1 |- (A.\x:al A) |= A

Syntax hints:  tv 1  *hb 3  kc 5  \kl 6   = ke 7  [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:  alimdv  184  alnex  186  ax5  207  ax7  209  ax10  213  axext  219