Theorem rcla4e 1919

Description: Restricted existential specialization, using implicit substitition.

Hypotheses

Ref	Expression
rcla4.1	⊢ (ψ → ∀xψ)
rcla4.2	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
rcla4e	⊢ ((A ∈ B ⋀ ψ) → ∃x ∈ B φ)

Distinct variable groups:   x,A   x,B

Proof of Theorem rcla4e

Step	Hyp	Ref	Expression
1		ax-17 1012	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2		ax-17 1012	. . . . 5 ⊢ (A ∈ B → ∀x A ∈ B)
3		rcla4.1	. . . . 5 ⊢ (ψ → ∀xψ)
4	2, 3	hban 1050	. . . 4 ⊢ ((A ∈ B ⋀ ψ) → ∀x(A ∈ B ⋀ ψ))
5		eleq1 1581	. . . . 5 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
6		rcla4.2	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
7	5, 6	anbi12d 639	. . . 4 ⊢ (x = A → ((x ∈ B ⋀ φ) ↔ (A ∈ B ⋀ ψ)))
8	1, 4, 7	cla4egf 1908	. . 3 ⊢ (A ∈ B → ((A ∈ B ⋀ ψ) → ∃x(x ∈ B ⋀ φ)))
9	8	anabsi5 506	. 2 ⊢ ((A ∈ B ⋀ ψ) → ∃x(x ∈ B ⋀ φ))
10		df-rex 1697	. 2 ⊢ (∃x ∈ B φ ↔ ∃x(x ∈ B ⋀ φ))
11	9, 10	sylibr 207	1 ⊢ ((A ∈ B ⋀ ψ) → ∃x ∈ B φ)

Syntax hints:   → wi 3   ↔ wb 153   ⋀ wa 230  ∀wal 995   = wceq 997   ∈ wcel 999  ∃wex 1021  ∃wrex 1693

This theorem is referenced by:  rcla4ev 1924  infcvgaux1i 7309  fgsb 10663  fgsb2 10668

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-ext 1504

This theorem depends on definitions:  df-bi 154  df-an 232  df-ex 1022  df-sb 1214  df-clab 1510  df-cleq 1515  df-clel 1518  df-rex 1697  df-v 1859

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