Theorem insti 114

Description: Instantiate a theorem with a new term. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypotheses

Ref	Expression
insti.1	⊢ C:α
insti.2	⊢ B:∗
insti.3	⊢ R⊧A
insti.4	⊢ ⊤⊧[(λx:α By:α) = B]
insti.5	⊢ [x:α = C]⊧[A = B]

Assertion

Ref	Expression
insti	⊢ R⊧B

Distinct variable groups:   x,y,R   y,B

Proof of Theorem insti

Step	Hyp	Ref	Expression
1		insti.3	. 2 ⊢ R⊧A
2		insti.4	. 2 ⊢ ⊤⊧[(λx:α By:α) = B]
3	1	ax-cb1 29	. . 3 ⊢ R:∗
4		wv 64	. . 3 ⊢ y:α:α
5	3, 4	ax-17 105	. 2 ⊢ ⊤⊧[(λx:α Ry:α) = R]
6		insti.5	. 2 ⊢ [x:α = C]⊧[A = B]
7	6	ax-cb1 29	. . 3 ⊢ [x:α = C]:∗
8	7, 3	eqid 83	. 2 ⊢ [x:α = C]⊧[R = R]
9	1, 2, 5, 6, 8	ax-inst 113	1 ⊢ R⊧B

This proof depends on syntax axioms:  tv 1  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12

This proof depends on axioms:  ax-syl 15  ax-jca 17  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wov 71  ax-17 105  ax-inst 113

This proof depends on definitions:  df-ov 73

This theorem is used by:  clf  115  exlimdv  167  cbvf  179  exlimd  183