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)]

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)