Theorem bj-equsalvwd

Description: Variant of equsalvw . (Contributed by BJ, 7-Oct-2024)

	Ref	Expression
Hypotheses	bj-equsalvwd.nf0	$$⊢\phi \to \forall x\phi$$
	bj-equsalvwd.nf	$$⊢\phi \to Ⅎ\text{'}x\chi$$
	bj-equsalvwd.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
Assertion	bj-equsalvwd	$$⊢\phi \to \left(\forall x\left(x=y\to \psi \right)\leftrightarrow \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-equsalvwd.nf0	$$⊢\phi \to \forall x\phi$$
2		bj-equsalvwd.nf	$$⊢\phi \to Ⅎ\text{'}x\chi$$
3		bj-equsalvwd.is	$$⊢\left(\phi \wedge x=y\right)\to \left(\psi \leftrightarrow \chi \right)$$
4	3	pm5.74da	$$⊢\phi \to \left(\left(x=y\to \psi \right)\leftrightarrow \left(x=y\to \chi \right)\right)$$
5	1 4	albidh	$$⊢\phi \to \left(\forall x\left(x=y\to \psi \right)\leftrightarrow \forall x\left(x=y\to \chi \right)\right)$$
6		bj-equsvt	$$⊢Ⅎ\text{'}x\chi \to \left(\forall x\left(x=y\to \chi \right)\leftrightarrow \chi \right)$$
7	2 6	syl	$$⊢\phi \to \left(\forall x\left(x=y\to \chi \right)\leftrightarrow \chi \right)$$
8	5 7	bitrd	$$⊢\phi \to \left(\forall x\left(x=y\to \psi \right)\leftrightarrow \chi \right)$$