Theorem elabgt 1886

Description: Membership in a class abstraction with implicit substitution. (Closed theorem version of elabg 1890.)

Assertion

Ref	Expression
elabgt	⊢ ((A ∈ B ⋀ ∀x(x = A → (φ ↔ ψ))) → (A ∈ {x∣φ} ↔ ψ))

Distinct variable groups:   x,A   ψ,x

Proof of Theorem elabgt

Step	Hyp	Ref	Expression
1		ax-17 968	. . . . . 6 ⊢ (y ∈ A → ∀x y ∈ A)
2		hbab1 1459	. . . . . 6 ⊢ (y ∈ {x∣φ} → ∀x y ∈ {x∣φ})
3	1, 2	hbel 1558	. . . . 5 ⊢ (A ∈ {x∣φ} → ∀x A ∈ {x∣φ})
4		ax-17 968	. . . . 5 ⊢ (ψ → ∀xψ)
5	3, 4	hbbi 1007	. . . 4 ⊢ ((A ∈ {x∣φ} ↔ ψ) → ∀x(A ∈ {x∣φ} ↔ ψ))
6	5	ax-gen 960	. . 3 ⊢ ∀x((A ∈ {x∣φ} ↔ ψ) → ∀x(A ∈ {x∣φ} ↔ ψ))
7		vtoclegft 1847	. . 3 ⊢ ((A ∈ B ⋀ ∀x((A ∈ {x∣φ} ↔ ψ) → ∀x(A ∈ {x∣φ} ↔ ψ)) ⋀ ∀x(x = A → (A ∈ {x∣φ} ↔ ψ))) → (A ∈ {x∣φ} ↔ ψ))
8	6, 7	mp3an2 901	. 2 ⊢ ((A ∈ B ⋀ ∀x(x = A → (A ∈ {x∣φ} ↔ ψ))) → (A ∈ {x∣φ} ↔ ψ))
9		eleq1 1526	. . . . . . 7 ⊢ (x = A → (x ∈ {x∣φ} ↔ A ∈ {x∣φ}))
10		abid 1458	. . . . . . 7 ⊢ (x ∈ {x∣φ} ↔ φ)
11	9, 10	syl5rbbr 533	. . . . . 6 ⊢ (x = A → (A ∈ {x∣φ} ↔ φ))
12	11	bibi1d 617	. . . . 5 ⊢ (x = A → ((A ∈ {x∣φ} ↔ ψ) ↔ (φ ↔ ψ)))
13	12	biimprd 154	. . . 4 ⊢ (x = A → ((φ ↔ ψ) → (A ∈ {x∣φ} ↔ ψ)))
14	13	a2i 9	. . 3 ⊢ ((x = A → (φ ↔ ψ)) → (x = A → (A ∈ {x∣φ} ↔ ψ)))
15	14	19.20i 989	. 2 ⊢ (∀x(x = A → (φ ↔ ψ)) → ∀x(x = A → (A ∈ {x∣φ} ↔ ψ)))
16	8, 15	sylan2 451	1 ⊢ ((A ∈ B ⋀ ∀x(x = A → (φ ↔ ψ))) → (A ∈ {x∣φ} ↔ ψ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 951   = wceq 953   ∈ wcel 955  {cab 1456

This theorem is referenced by:  sbcel12g 2001  sbceqdig 2002

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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