New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  raleqi GIF version

Theorem raleqi 2811
 Description: Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypothesis
Ref Expression
raleq1i.1 A = B
Assertion
Ref Expression
raleqi (x A φx B φ)
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem raleqi
StepHypRef Expression
1 raleq1i.1 . 2 A = B
2 raleq 2807 . 2 (A = B → (x A φx B φ))
31, 2ax-mp 5 1 (x A φx B φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642  ∀wral 2614 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-ral 2619 This theorem is referenced by:  ralrab2  3002  ralprg  3775  raltpg  3777  ssofss  4076  ralxp  4825
 Copyright terms: Public domain W3C validator