Theorem hbth 109

Description: Hypothesis builder for a theorem. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypotheses

Ref	Expression
hbth.1	|- B:al
hbth.2	|- R |= A

Assertion

Ref	Expression
hbth	|- R |= [(\x:al AB) = A]

Distinct variable group:   x,R

Proof of Theorem hbth

Step	Hyp	Ref	Expression
1		hbth.2	. . 3 |- R |= A
2	1	ax-cb2 30	. 2 |- A:*
3		hbth.1	. 2 |- B:al
4		wtru 43	. . 3 |- T.:*
5	1	eqtru 86	. . 3 |- R |= [T. = A]
6	4, 5	eqcomi 79	. 2 |- R |= [A = T.]
7	1	ax-cb1 29	. . 3 |- R:*
8	4, 3, 7	a17i 106	. 2 |- R |= [(\x:al T.B) = T.]
9	2, 3, 6, 8	hbxfr 108	1 |- R |= [(\x:al AB) = A]

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

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-wl 65  ax-leq 69  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-17 105

This theorem depends on definitions:  df-ov 73

This theorem is referenced by:  ax4g  149