Theorem ax8 198

Description: Axiom of Equality. Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105.

Hypotheses

Ref	Expression
ax8.1	|- A:al
ax8.2	|- B:al
ax8.3	|- C:al

Assertion

Ref	Expression
ax8	|- T. |= [[A = B] ==> [[A = C] ==> [B = C]]]

Proof of Theorem ax8

Step	Hyp	Ref	Expression
1		ax8.2	. . . . 5 |- B:al
2		ax8.1	. . . . . 6 |- A:al
3	2, 1	weqi 68	. . . . . . 7 |- [A = B]:*
4		ax8.3	. . . . . . . 8 |- C:al
5	2, 4	weqi 68	. . . . . . 7 |- [A = C]:*
6	3, 5	simpl 22	. . . . . 6 |- ([A = B], [A = C]) |= [A = B]
7	2, 6	eqcomi 70	. . . . 5 |- ([A = B], [A = C]) |= [B = A]
8	3, 5	simpr 23	. . . . 5 |- ([A = B], [A = C]) |= [A = C]
9	1, 7, 8	eqtri 85	. . . 4 |- ([A = B], [A = C]) |= [B = C]
10	9	ex 148	. . 3 |- [A = B] |= [[A = C] ==> [B = C]]
11		wtru 40	. . 3 |- T.:*
12	10, 11	adantl 51	. 2 |- (T., [A = B]) |= [[A = C] ==> [B = C]]
13	12	ex 148	1 |- T. |= [[A = B] ==> [[A = C] ==> [B = C]]]

Syntax hints:   = ke 7  T.kt 8  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 111

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-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103

This theorem depends on definitions:  df-ov 65  df-an 118  df-im 119

This theorem is referenced by: (None)