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	⊢ Ⅎ 𝑥 𝜓
	bj-vtoclg1f1.maj	⊢ ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) )
	bj-vtoclg1f1.min	⊢ 𝜑
Assertion	bj-vtoclg1f1	⊢ ( ∃ 𝑦 𝑦 = 𝐴 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-vtoclg1f1.nf	⊢ Ⅎ 𝑥 𝜓
2		bj-vtoclg1f1.maj	⊢ ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) )
3		bj-vtoclg1f1.min	⊢ 𝜑
4		bj-denotes	⊢ ( ∃ 𝑦 𝑦 = 𝐴 ↔ ∃ 𝑥 𝑥 = 𝐴 )
5	1 2 3	bj-exlimmpi	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 )
6	4 5	sylbi	⊢ ( ∃ 𝑦 𝑦 = 𝐴 → 𝜓 )