Theorem rexeq1f 1787

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions.

Hypotheses

Ref	Expression
raleq1f.1	⊢ (y ∈ A → ∀x y ∈ A)
raleq1f.2	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

Ref	Expression
rexeq1f	⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))

Distinct variable groups:   y,A   y,B   x,y

Proof of Theorem rexeq1f

Step	Hyp	Ref	Expression
1		raleq1f.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2		raleq1f.2	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
3	1, 2	hbeq 1568	. . 3 ⊢ (A = B → ∀x A = B)
4		eleq2 1538	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
5	4	anbi1d 619	. . 3 ⊢ (A = B → ((x ∈ A ⋀ φ) ↔ (x ∈ B ⋀ φ)))
6	3, 5	exbid 1107	. 2 ⊢ (A = B → (∃x(x ∈ A ⋀ φ) ↔ ∃x(x ∈ B ⋀ φ)))
7		df-rex 1653	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ⋀ φ))
8		df-rex 1653	. 2 ⊢ (∃x ∈ B φ ↔ ∃x(x ∈ B ⋀ φ))
9	6, 7, 8	3bitr4g 557	1 ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 956   = wceq 958   ∈ wcel 960  ∃wex 982  ∃wrex 1649

This theorem is referenced by:  rexeq1 1790  zfrep6 3620

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-cleq 1472  df-clel 1475  df-rex 1653

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