Theorem 19.8a 170

Description: Existential introduction. (Contributed by Mario Carneiro, 9-Oct-2014.)

Hypothesis

Ref	Expression
19.8a.1	|- A:*

Assertion

Ref	Expression
19.8a	|- A |= (E.\x:al A)

Proof of Theorem 19.8a

Step	Hyp	Ref	Expression
1		19.8a.1	. . . 4 |- A:*
2	1	ax-id 24	. . 3 |- A |= A
3	1	beta 92	. . . 4 |- T. |= [(\x:al Ax:al) = A]
4	1, 3	a1i 28	. . 3 |- A |= [(\x:al Ax:al) = A]
5	2, 4	mpbir 87	. 2 |- A |= (\x:al Ax:al)
6	1	wl 66	. . 3 |- \x:al A:(al -> *)
7		wv 64	. . 3 |- x:al:al
8	6, 7	ax4e 168	. 2 |- (\x:al Ax:al) |= (E.\x:al A)
9	5, 8	syl 16	1 |- A |= (E.\x:al A)

Syntax hints:  tv 1  *hb 3  kc 5  \kl 6   = ke 7  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  E.tex 123

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-distrl 70  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-an 128  df-im 129  df-ex 131

This theorem is referenced by:  eximdv  185  alnex  186  ax9  212