ax17 - Higher-Order Logic Explorer

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Theorem ax17 205

Description: Axiom to quantify a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

**Hypothesis**
Ref	Expression
ax17.1	⊢ A:∗

**Assertion**
Ref	Expression
ax17	⊢ ⊤⊧[A ⇒ (∀λx:α A)]

Distinct variable groups: x,A α,x

Proof of Theorem ax17 Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax17.1	. 2 ⊢ A:∗
2		wv 58	. . 3 ⊢ y:α:α
3	1, 2	ax-17 95	. 2 ⊢ ⊤⊧[(λx:α Ay:α) = A]
4	1, 3	isfree 176	1 ⊢ ⊤⊧[A ⇒ (∀λx:α A)]

Colors of variables: type var term

Syntax hints: tv 1 ∗hb 3 kc 5 λkl 6 ⊤kt 8 [kbr 9 ⊧wffMMJ2 11 wffMMJ2t 12 ⇒ tim 111 ∀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-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 df-an 118 df-im 119

This theorem is referenced by: (None)

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