Theorem elabgf 1416

Description: Membership in a class abstraction with implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound variable hypotheses in place of distinct variable restrictions.

Hypotheses

Ref	Expression
elabgf.1	⊢ (y ∈ A → ∀x y ∈ A)
elabgf.2	⊢ (ψ → ∀xψ)
elabgf.3	⊢ (x = A → (φ ↔ ψ))

<R>

Assertion

Ref	Expression
elabgf	⊢ (A ∈ B → (A ∈ {x∣φ} ↔ ψ))

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

Proof of Theorem elabgf

Step	Hyp	Ref	Expression
1		elabgf.1	. 2 ⊢ (y ∈ A → ∀x y ∈ A)
2		hbab1 1095	. . . 4 ⊢ (z ∈ {x∣φ} → ∀x z ∈ {x∣φ})
3	1, 2	hbel 1172	. . 3 ⊢ (A ∈ {x∣φ} → ∀x A ∈ {x∣φ})
4		elabgf.2	. . 3 ⊢ (ψ → ∀xψ)
5	3, 4	hbbi 705	. 2 ⊢ ((A ∈ {x∣φ} ↔ ψ) → ∀x(A ∈ {x∣φ} ↔ ψ))
6		eleq1 1149	. . 3 ⊢ (x = A → (x ∈ {x∣φ} ↔ A ∈ {x∣φ}))
7		elabgf.3	. . 3 ⊢ (x = A → (φ ↔ ψ))
8	6, 7	bibi12d 477	. 2 ⊢ (x = A → ((x ∈ {x∣φ} ↔ φ) ↔ (A ∈ {x∣φ} ↔ ψ)))
9		abid 1094	. 2 ⊢ (x ∈ {x∣φ} ↔ φ)
10	1, 5, 8, 9	vtoclgf 1382	1 ⊢ (A ∈ B → (A ∈ {x∣φ} ↔ ψ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  {cab 1090   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  elabg 1417  elrabf 1421  cardprc 3667

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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