Theorem rexeqf 2805

Description: Equality theorem for restricted existential quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 9-Oct-2003.) (Revised by Andrew Salmon, 11-Jul-2011.)

Hypotheses

Ref	Expression
raleq1f.1	⊢ ℲxA
raleq1f.2	⊢ ℲxB

Assertion

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

Proof of Theorem rexeqf

Step	Hyp	Ref	Expression
1		raleq1f.1	. . . 4 ⊢ ℲxA
2		raleq1f.2	. . . 4 ⊢ ℲxB
3	1, 2	nfeq 2497	. . 3 ⊢ Ⅎx A = B
4		eleq2 2414	. . . 4 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
5	4	anbi1d 685	. . 3 ⊢ (A = B → ((x ∈ A ∧ φ) ↔ (x ∈ B ∧ φ)))
6	3, 5	exbid 1773	. 2 ⊢ (A = B → (∃x(x ∈ A ∧ φ) ↔ ∃x(x ∈ B ∧ φ)))
7		df-rex 2621	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
8		df-rex 2621	. 2 ⊢ (∃x ∈ B φ ↔ ∃x(x ∈ B ∧ φ))
9	6, 7, 8	3bitr4g 279	1 ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  ∃wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621

This theorem is referenced by:  rexeq  2809