Theorem pm14.122b

Description: Theorem *14.122 in WhiteheadRussell p. 185. (Contributed by Andrew Salmon, 9-Jun-2011)

	Ref	Expression
Assertion	pm14.122b	$$⊢A\in V\to \left(\left(\forall x\left(\phi \to x=A\right)\wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\leftrightarrow \left(\forall x\left(\phi \to x=A\right)\wedge \exists x\phi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		eqeq2	$$⊢y=A\to \left(x=y\leftrightarrow x=A\right)$$
2	1	imbi2d	$$⊢y=A\to \left(\left(\phi \to x=y\right)\leftrightarrow \left(\phi \to x=A\right)\right)$$
3	2	albidv	$$⊢y=A\to \left(\forall x\left(\phi \to x=y\right)\leftrightarrow \forall x\left(\phi \to x=A\right)\right)$$
4		dfsbcq	$$⊢y=A\to \left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)$$
5	4	bibi1d	$$⊢y=A\to \left(\left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \leftrightarrow \exists x\phi \right)\leftrightarrow \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \exists x\phi \right)\right)$$
6	3 5	imbi12d	$$⊢y=A\to \left(\left(\forall x\left(\phi \to x=y\right)\to \left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \leftrightarrow \exists x\phi \right)\right)\leftrightarrow \left(\forall x\left(\phi \to x=A\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \exists x\phi \right)\right)\right)$$
7		sbc5	$$⊢\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \leftrightarrow \exists x\left(x=y\wedge \phi \right)$$
8		nfa1	$$⊢Ⅎx\forall x\left(\phi \to x=y\right)$$
9		simpr	$$⊢\left(x=y\wedge \phi \right)\to \phi$$
10		ancr	$$⊢\left(\phi \to x=y\right)\to \left(\phi \to \left(x=y\wedge \phi \right)\right)$$
11	10	sps	$$⊢\forall x\left(\phi \to x=y\right)\to \left(\phi \to \left(x=y\wedge \phi \right)\right)$$
12	9 11	impbid2	$$⊢\forall x\left(\phi \to x=y\right)\to \left(\left(x=y\wedge \phi \right)\leftrightarrow \phi \right)$$
13	8 12	exbid	$$⊢\forall x\left(\phi \to x=y\right)\to \left(\exists x\left(x=y\wedge \phi \right)\leftrightarrow \exists x\phi \right)$$
14	7 13	syl5bb	$$⊢\forall x\left(\phi \to x=y\right)\to \left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \leftrightarrow \exists x\phi \right)$$
15	6 14	vtoclg	$$⊢A\in V\to \left(\forall x\left(\phi \to x=A\right)\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \exists x\phi \right)\right)$$
16	15	pm5.32d	$$⊢A\in V\to \left(\left(\forall x\left(\phi \to x=A\right)\wedge \underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \right)\leftrightarrow \left(\forall x\left(\phi \to x=A\right)\wedge \exists x\phi \right)\right)$$