This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem vtocld

Description: Implicit substitution of a class for a setvar variable. (Contributed by Mario Carneiro, 15-Oct-2016) Avoid ax-10 , ax-11 , ax-12 . (Revised by SN, 2-Sep-2024)

	Ref	Expression
Hypotheses	vtocld.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	vtocld.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
	vtocld.3	⊢ ( 𝜑 → 𝜓 )
Assertion	vtocld	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		vtocld.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		vtocld.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3		vtocld.3	⊢ ( 𝜑 → 𝜓 )
4		elisset	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
5	1 4	syl	⊢ ( 𝜑 → ∃ 𝑥 𝑥 = 𝐴 )
6	3	adantr	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝜓 )
7	6 2	mpbid	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → 𝜒 )
8	5 7	exlimddv	⊢ ( 𝜑 → 𝜒 )