Theorem rexeqi 2813

Description: Equality inference for restricted existential qualifier. (Contributed by Mario Carneiro, 23-Apr-2015.)

Hypothesis

Ref	Expression
raleq1i.1	⊢ A = B

Assertion

Ref	Expression
rexeqi	⊢ (∃x ∈ A φ ↔ ∃x ∈ B φ)

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   φ(x)

Proof of Theorem rexeqi

Step	Hyp	Ref	Expression
1		raleq1i.1	. 2 ⊢ A = B
2		rexeq 2809	. 2 ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))
3	1, 2	ax-mp 5	1 ⊢ (∃x ∈ A φ ↔ ∃x ∈ B φ)

Syntax hints:   ↔ wb 176   = wceq 1642  ∃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:  rexrab2  3005  rexprg  3777  rextpg  3779  opeq  4620  rexxp  4827  clos1basesucg  5885