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Metamath Proof Explorer

Theorem vtocl2gf

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 25-Apr-1995)

Ref Expression
Hypotheses vtocl2gf.1 ⊢ Ⅎ 𝑥 𝐴
vtocl2gf.2 ⊢ Ⅎ 𝑦 𝐴
vtocl2gf.3 ⊢ Ⅎ 𝑦 𝐵
vtocl2gf.4 ⊢ Ⅎ 𝑥 𝜓
vtocl2gf.5 ⊢ Ⅎ 𝑦 𝜒
vtocl2gf.6 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
vtocl2gf.7 ⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
vtocl2gf.8 ⊢ 𝜑
Assertion vtocl2gf ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝜒 )

Proof

Step Hyp Ref Expression
1 vtocl2gf.1 ⊢ Ⅎ 𝑥 𝐴
2 vtocl2gf.2 ⊢ Ⅎ 𝑦 𝐴
3 vtocl2gf.3 ⊢ Ⅎ 𝑦 𝐵
4 vtocl2gf.4 ⊢ Ⅎ 𝑥 𝜓
5 vtocl2gf.5 ⊢ Ⅎ 𝑦 𝜒
6 vtocl2gf.6 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
7 vtocl2gf.7 ⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
8 vtocl2gf.8 ⊢ 𝜑
9 elex ⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ V )
10 2 nfel1 ⊢ Ⅎ 𝑦 𝐴 ∈ V
11 10 5 nfim ⊢ Ⅎ 𝑦 ( 𝐴 ∈ V → 𝜒 )
12 7 imbi2d ⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ∈ V → 𝜓 ) ↔ ( 𝐴 ∈ V → 𝜒 ) ) )
13 1 4 6 8 vtoclgf ⊢ ( 𝐴 ∈ V → 𝜓 )
14 3 11 12 13 vtoclgf ⊢ ( 𝐵 ∈ 𝑊 → ( 𝐴 ∈ V → 𝜒 ) )
15 9 14 mpan9 ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝜒 )