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	|- F/ x ps
	bj-vtoclg1f1.maj	|- ( x = A -> ( ph -> ps ) )
	bj-vtoclg1f1.min	|- ph
Assertion	bj-vtoclg1f1	|- ( E. y y = A -> ps )

Proof

Step	Hyp	Ref	Expression
1		bj-vtoclg1f1.nf	|- F/ x ps
2		bj-vtoclg1f1.maj	|- ( x = A -> ( ph -> ps ) )
3		bj-vtoclg1f1.min	|- ph
4		bj-denotes	|- ( E. y y = A <-> E. x x = A )
5	1 2 3	bj-exlimmpi	|- ( E. x x = A -> ps )
6	4 5	sylbi	|- ( E. y y = A -> ps )