Theorem spsbc

Description: Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of Quine p. 44. This is Frege's ninth axiom per Proposition 58 of Frege1879 p. 51. See also stdpc4 and rspsbc . (Contributed by NM, 16-Jan-2004)

	Ref	Expression
Assertion	spsbc	$$⊢A\in V\to \left(\forall x\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		stdpc4	$$⊢\forall x\phi \to \left[y/x\right]\phi$$
2		sbsbc	$$⊢\left[y/x\right]\phi \leftrightarrow \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi$$
3	1 2	sylib	$$⊢\forall x\phi \to \underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi$$
4		dfsbcq	$$⊢y=A\to \left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
5	3 4	imbitrid	$$⊢y=A\to \left(\forall x\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
6	5	vtocleg	$$⊢A\in V\to \left(\forall x\phi \to \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$