Metamath Proof Explorer

Theorem rspct

Description: A closed version of rspc . (Contributed by Andrew Salmon, 6-Jun-2011)

Ref Expression
Hypothesis rspct.1 ⊢Ⅎxψ
Assertion rspct ⊢∀xx=A→φ↔ψ→A∈B→∀x∈Bφ→ψ

Proof

Step Hyp Ref Expression
1 rspct.1 ⊢Ⅎxψ
2 df-ral ⊢∀x∈Bφ↔∀xx∈B→φ
3 eleq1 ⊢x=A→x∈B↔A∈B
4 3 adantr ⊢x=A∧φ↔ψ→x∈B↔A∈B
5 simpr ⊢x=A∧φ↔ψ→φ↔ψ
6 4 5 imbi12d ⊢x=A∧φ↔ψ→x∈B→φ↔A∈B→ψ
7 6 ex ⊢x=A→φ↔ψ→x∈B→φ↔A∈B→ψ
8 7 a2i ⊢x=A→φ↔ψ→x=A→x∈B→φ↔A∈B→ψ
9 8 alimi ⊢∀xx=A→φ↔ψ→∀xx=A→x∈B→φ↔A∈B→ψ
10 nfv ⊢ℲxA∈B
11 10 1 nfim ⊢ℲxA∈B→ψ
12 nfcv ⊢Ⅎ_xA
13 11 12 spcgft ⊢∀xx=A→x∈B→φ↔A∈B→ψ→A∈B→∀xx∈B→φ→A∈B→ψ
14 9 13 syl ⊢∀xx=A→φ↔ψ→A∈B→∀xx∈B→φ→A∈B→ψ
15 2 14 syl7bi ⊢∀xx=A→φ↔ψ→A∈B→∀x∈Bφ→A∈B→ψ
16 15 com34 ⊢∀xx=A→φ↔ψ→A∈B→A∈B→∀x∈Bφ→ψ
17 16 pm2.43d ⊢∀xx=A→φ↔ψ→A∈B→∀x∈Bφ→ψ