Theorem bj-vtoclg1f1

Description: The FOL content of vtoclg1f (hence not using ax-ext , df-cleq , df-nfc , df-v ). Note the weakened "major" hypothesis and the disjoint variable condition between x and A (needed since the nonfreeness quantifier for classes is not available without ax-ext ; as a byproduct, this dispenses with ax-11 and ax-13 ). (Contributed by BJ, 30-Apr-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-vtoclg1f1.nf	$$⊢Ⅎx\psi$$
	bj-vtoclg1f1.maj	$$⊢x=A\to \left(\phi \to \psi \right)$$
	bj-vtoclg1f1.min	$$⊢\phi$$
Assertion	bj-vtoclg1f1	$$⊢\exists yy=A\to \psi$$

Proof

Step	Hyp	Ref	Expression
1		bj-vtoclg1f1.nf	$$⊢Ⅎx\psi$$
2		bj-vtoclg1f1.maj	$$⊢x=A\to \left(\phi \to \psi \right)$$
3		bj-vtoclg1f1.min	$$⊢\phi$$
4		iseqsetv-clel	$$⊢\exists yy=A\leftrightarrow \exists xx=A$$
5	1 2 3	bj-exlimmpi	$$⊢\exists xx=A\to \psi$$
6	4 5	sylbi	$$⊢\exists yy=A\to \psi$$