Theorem ceqsalgALT

Description: Alternate proof of ceqsalg , not using ceqsalt . (Contributed by NM, 29-Oct-2003) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Revised by BJ, 29-Sep-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ceqsalg.1	$$⊢Ⅎx\psi$$
	ceqsalg.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	ceqsalgALT	$$⊢A\in V\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ceqsalg.1	$$⊢Ⅎx\psi$$
2		ceqsalg.2	$$⊢x=A\to \left(\phi \leftrightarrow \psi \right)$$
3		elisset	$$⊢A\in V\to \exists xx=A$$
4		nfa1	$$⊢Ⅎx\forall x\left(x=A\to \phi \right)$$
5	2	biimpd	$$⊢x=A\to \left(\phi \to \psi \right)$$
6	5	a2i	$$⊢\left(x=A\to \phi \right)\to \left(x=A\to \psi \right)$$
7	6	sps	$$⊢\forall x\left(x=A\to \phi \right)\to \left(x=A\to \psi \right)$$
8	4 1 7	exlimd	$$⊢\forall x\left(x=A\to \phi \right)\to \left(\exists xx=A\to \psi \right)$$
9	3 8	syl5com	$$⊢A\in V\to \left(\forall x\left(x=A\to \phi \right)\to \psi \right)$$
10	2	biimprcd	$$⊢\psi \to \left(x=A\to \phi \right)$$
11	1 10	alrimi	$$⊢\psi \to \forall x\left(x=A\to \phi \right)$$
12	9 11	impbid1	$$⊢A\in V\to \left(\forall x\left(x=A\to \phi \right)\leftrightarrow \psi \right)$$