Theorem cla4ev 169

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

Hypotheses

Ref	Expression
cla4ev.1	|- A:*
cla4ev.2	|- B:al
cla4ev.3	|- [x:al = B] |= [A = C]

Assertion

Ref	Expression
cla4ev	|- C |= (E.\x:al A)

Distinct variable groups:   x,B   x,C   al,x

Proof of Theorem cla4ev

Step	Hyp	Ref	Expression
1		cla4ev.1	. . . . 5 |- A:*
2		cla4ev.3	. . . . 5 |- [x:al = B] |= [A = C]
3	1, 2	eqtypi 78	. . . 4 |- C:*
4	3	id 25	. . 3 |- C |= C
5		cla4ev.2	. . . . 5 |- B:al
6	1, 5, 2	cl 116	. . . 4 |- T. |= [(\x:al AB) = C]
7	3, 6	a1i 28	. . 3 |- C |= [(\x:al AB) = C]
8	4, 7	mpbir 87	. 2 |- C |= (\x:al AB)
9	1	wl 66	. . 3 |- \x:al A:(al -> *)
10	9, 5	ax4e 168	. 2 |- (\x:al AB) |= (E.\x:al A)
11	8, 10	syl 16	1 |- C |= (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:  axpow  221  axun  222