Theorem elab3g 1420

Description: Membership in a class abstraction using implicit substitution.

Hypotheses

Ref	Expression
elab3g.1	⊢ (ψ → A ∈ V)
elab3g.2	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
elab3g	⊢ (A ∈ {x∣φ} ↔ ψ)

Distinct variable group(s):   ψ,x   x,A

Proof of Theorem elab3g

Step	Hyp	Ref	Expression
1		elab3g.2	. . . 4 ⊢ (x = A → (φ ↔ ψ))
2	1	elabg 1417	. . 3 ⊢ (A ∈ {x∣φ} → (A ∈ {x∣φ} ↔ ψ))
3	2	ibi 449	. 2 ⊢ (A ∈ {x∣φ} → ψ)
4		elab3g.1	. . . . 5 ⊢ (ψ → A ∈ V)
5	1	elabg 1417	. . . . 5 ⊢ (A ∈ V → (A ∈ {x∣φ} ↔ ψ))
6	4, 5	syl 12	. . . 4 ⊢ (ψ → (A ∈ {x∣φ} ↔ ψ))
7	6	biimprd 136	. . 3 ⊢ (ψ → (ψ → A ∈ {x∣φ}))
8	7	pm2.43i 58	. 2 ⊢ (ψ → A ∈ {x∣φ})
9	3, 8	impbi 139	1 ⊢ (A ∈ {x∣φ} ↔ ψ)

Syntax hints:   → wi 2   ↔ wb 127  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  elmap 3265  elq 4629  hosmvalt 5487  hodmvalt 5488

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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