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Metamath Proof Explorer

Theorem vtoclg1f

Description: Version of vtoclgf with one nonfreeness hypothesis replaced with a disjoint variable condition, thus avoiding dependency on ax-10 and ax-11 . (Contributed by BJ, 1-May-2019)

	Ref	Expression
Hypotheses	vtoclg1f.nf	⊢ Ⅎ 𝑥 𝜓
	vtoclg1f.maj	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	vtoclg1f.min	⊢ 𝜑
Assertion	vtoclg1f	⊢ ( 𝐴 ∈ 𝑉 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		vtoclg1f.nf	⊢ Ⅎ 𝑥 𝜓
2		vtoclg1f.maj	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		vtoclg1f.min	⊢ 𝜑
4		elisset	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
5	3 2	mpbii	⊢ ( 𝑥 = 𝐴 → 𝜓 )
6	1 5	exlimi	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 )
7	4 6	syl	⊢ ( 𝐴 ∈ 𝑉 → 𝜓 )