Theorem rexeqf 1322

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

Hypotheses

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

Assertion

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

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

Proof of Theorem rexeqf

Step	Hyp	Ref	Expression
1		raleqf.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2		raleqf.2	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
3	1, 2	hbeq 1171	. . 3 ⊢ (A = B → ∀x A = B)
4		eleq2 1150	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
5	4	anbi1d 469	. . 3 ⊢ (A = B → ((x ∈ A ∧ φ) ↔ (x ∈ B ∧ φ)))
6	3, 5	biexd 783	. 2 ⊢ (A = B → (∃x(x ∈ A ∧ φ) ↔ ∃x(x ∈ B ∧ φ)))
7		df-rex 1206	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
8		df-rex 1206	. 2 ⊢ (∃x ∈ B φ ↔ ∃x(x ∈ B ∧ φ))
9	6, 7, 8	3bitr4g 428	1 ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  rexeq 1325  zfrep6 2744

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-9 799  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-rex 1206

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