Theorem alrimiv 141

Description: If one can prove R |= A where R does not contain x, then A is true for all x.

Hypothesis

Ref	Expression
alrimiv.1	|- R |= A

Assertion

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

Distinct variable groups:   x,R   al,x

Proof of Theorem alrimiv

Step	Hyp	Ref	Expression
1		alrimiv.1	. . . 4 |- R |= A
2	1	ax-cb2 30	. . 3 |- A:*
3		wtru 40	. . . 4 |- T.:*
4	1	eqtru 76	. . . 4 |- R |= [T. = A]
5	3, 4	eqcomi 70	. . 3 |- R |= [A = T.]
6	2, 5	leq 81	. 2 |- R |= [\x:al A = \x:al T.]
7	1	ax-cb1 29	. . 3 |- R:*
8	2	wl 59	. . . 4 |- \x:al A:(al -> *)
9	8	alval 132	. . 3 |- T. |= [(A.\x:al A) = [\x:al A = \x:al T.]]
10	7, 9	a1i 28	. 2 |- R |= [(A.\x:al A) = [\x:al A = \x:al T.]]
11	6, 10	mpbir 77	1 |- R |= (A.\x:al A)

Syntax hints:  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  A.tal 112

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-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-hbl1 93  ax-17 95  ax-inst 103

This theorem depends on definitions:  df-ov 65  df-al 116

This theorem is referenced by:  exlimdv2  156  ax4e  158  exlimd  171  axgen  197  ax10  200  ax11  201  axrep  207  axpow  208  axun  209