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Theorem raleqf 2803
 Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
raleq1f.1 xA
raleq1f.2 xB
Assertion
Ref Expression
raleqf (A = B → (x A φx B φ))

Proof of Theorem raleqf
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))
54imbi1d 308 . . 3 (A = B → ((x Aφ) ↔ (x Bφ)))
63, 5albid 1772 . 2 (A = B → (x(x Aφ) ↔ x(x Bφ)))
7 df-ral 2619 . 2 (x A φx(x Aφ))
8 df-ral 2619 . 2 (x B φx(x Bφ))
96, 7, 83bitr4g 279 1 (A = B → (x A φx B φ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  ∀wral 2614 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-ral 2619 This theorem is referenced by:  raleq  2807
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