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Theorem rexeqf 2804
 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 xA
raleq1f.2 xB
Assertion
Ref Expression
rexeqf (A = B → (x A φx B φ))

Proof of Theorem rexeqf
StepHypRef Expression
1 raleq1f.1 . . . 4 xA
2 raleq1f.2 . . . 4 xB
31, 2nfeq 2496 . . 3 x A = B
4 eleq2 2414 . . . 4 (A = B → (x Ax B))
54anbi1d 685 . . 3 (A = B → ((x A φ) ↔ (x B φ)))
63, 5exbid 1773 . 2 (A = B → (x(x A φ) ↔ x(x B φ)))
7 df-rex 2620 . 2 (x A φx(x A φ))
8 df-rex 2620 . 2 (x B φx(x B φ))
96, 7, 83bitr4g 279 1 (A = B → (x A φx B φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  ∃wrex 2615 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 2478  df-rex 2620 This theorem is referenced by:  rexeq  2808
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