cbv - Higher-Order Logic Explorer

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Theorem cbv 168

Description: Change bound variables in a lambda abstraction.

**Hypotheses**
Ref	Expression
cbv.1	⊢ A:β
cbv.2	⊢ [x:α = y:α]⊧[A = B]

**Assertion**
Ref	Expression
cbv	⊢ ⊤⊧[λx:α A = λy:α B]

Distinct variable groups: y,A x,B x,y,α β,y

Proof of Theorem cbv Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbv.1	. 2 ⊢ A:β
2		wv 58	. . 3 ⊢ z:α:α
3	1, 2	ax-17 95	. 2 ⊢ ⊤⊧[(λy:α Az:α) = A]
4		cbv.2	. . . 4 ⊢ [x:α = y:α]⊧[A = B]
5	1, 4	eqtypi 69	. . 3 ⊢ B:β
6	5, 2	ax-17 95	. 2 ⊢ ⊤⊧[(λx:α Bz:α) = B]
7	1, 3, 6, 4	cbvf 167	1 ⊢ ⊤⊧[λx:α A = λy:α B]

Colors of variables: type var term

Syntax hints: tv 1 λkl 6 = ke 7 ⊤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-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 ax-eta 165

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

This theorem is referenced by: ax10 200