Theorem rexeq 2809

Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 29-Oct-1995.)

Assertion

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

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   φ(x)

Proof of Theorem rexeq

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲxA
2		nfcv 2490	. 2 ⊢ ℲxB
3	1, 2	rexeqf 2805	1 ⊢ (A = B → (∃x ∈ A φ ↔ ∃x ∈ B φ))

Syntax hints:   → wi 4   ↔ 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:  rexeqi  2813  rexeqdv  2815  rexeqbi1dv  2817  unieq  3901  xpkeq1  4199  xpkeq2  4200  imakeq2  4226  tfineq  4489  nnadjoin  4521  tfinnn  4535  imaeq2  4939  qseq1  5975  brlecg  6113  ovmuc  6131  tceq  6159  lec0cg  6199  sbth  6207  dflec3  6222  lenc  6224