Theorem bj-gabeqd

Description: Equality of generalized class abstractions. Deduction form. (Contributed by BJ, 4-Oct-2024)

	Ref	Expression
Hypotheses	bj-gabeqd.nf	$$⊢\phi \to \forall x\phi$$
	bj-gabeqd.c	$$⊢\phi \to A=B$$
	bj-gabeqd.f	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	bj-gabeqd	$$⊢\phi \to {{A\left|x\right|\psi }}={{B\left|x\right|\chi }}$$

Proof

Step	Hyp	Ref	Expression
1		bj-gabeqd.nf	$$⊢\phi \to \forall x\phi$$
2		bj-gabeqd.c	$$⊢\phi \to A=B$$
3		bj-gabeqd.f	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
4	3	biimpd	$$⊢\phi \to \left(\psi \to \chi \right)$$
5	1 2 4	bj-gabssd	$$⊢\phi \to {{A\left|x\right|\psi }}\subseteq {{B\left|x\right|\chi }}$$
6	2	eqcomd	$$⊢\phi \to B=A$$
7	3	biimprd	$$⊢\phi \to \left(\chi \to \psi \right)$$
8	1 6 7	bj-gabssd	$$⊢\phi \to {{B\left|x\right|\chi }}\subseteq {{A\left|x\right|\psi }}$$
9	5 8	eqssd	$$⊢\phi \to {{A\left|x\right|\psi }}={{B\left|x\right|\chi }}$$