Theorem raleqf

Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See raleq for a version based on fewer axioms. (Contributed by NM, 7-Mar-2004) (Revised by Andrew Salmon, 11-Jul-2011)

	Ref	Expression
Hypotheses	raleqf.1	$$⊢\underset{\_}{Ⅎ}xA$$
	raleqf.2	$$⊢\underset{\_}{Ⅎ}xB$$
Assertion	raleqf	$$⊢A=B\to \left(\forall x\in A\phi \leftrightarrow \forall x\in B\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		raleqf.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		raleqf.2	$$⊢\underset{\_}{Ⅎ}xB$$
3	1 2	nfeq	$$⊢ℲxA=B$$
4		eleq2	$$⊢A=B\to \left(x\in A\leftrightarrow x\in B\right)$$
5	4	imbi1d	$$⊢A=B\to \left(\left(x\in A\to \phi \right)\leftrightarrow \left(x\in B\to \phi \right)\right)$$
6	3 5	albid	$$⊢A=B\to \left(\forall x\left(x\in A\to \phi \right)\leftrightarrow \forall x\left(x\in B\to \phi \right)\right)$$
7		df-ral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$
8		df-ral	$$⊢\forall x\in B\phi \leftrightarrow \forall x\left(x\in B\to \phi \right)$$
9	6 7 8	3bitr4g	$$⊢A=B\to \left(\forall x\in A\phi \leftrightarrow \forall x\in B\phi \right)$$