ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ceqsrexv Structured version   GIF version

Theorem ceqsrexv 2668
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 30-Apr-2004.)
Hypothesis
Ref Expression
ceqsrexv.1 (x = A → (φψ))
Assertion
Ref Expression
ceqsrexv (A B → (x B (x = A φ) ↔ ψ))
Distinct variable groups:   x,A   x,B   ψ,x
Allowed substitution hint:   φ(x)

Proof of Theorem ceqsrexv
StepHypRef Expression
1 df-rex 2306 . . 3 (x B (x = A φ) ↔ x(x B (x = A φ)))
2 an12 495 . . . 4 ((x = A (x B φ)) ↔ (x B (x = A φ)))
32exbii 1493 . . 3 (x(x = A (x B φ)) ↔ x(x B (x = A φ)))
41, 3bitr4i 176 . 2 (x B (x = A φ) ↔ x(x = A (x B φ)))
5 eleq1 2097 . . . . 5 (x = A → (x BA B))
6 ceqsrexv.1 . . . . 5 (x = A → (φψ))
75, 6anbi12d 442 . . . 4 (x = A → ((x B φ) ↔ (A B ψ)))
87ceqsexgv 2667 . . 3 (A B → (x(x = A (x B φ)) ↔ (A B ψ)))
98bianabs 543 . 2 (A B → (x(x = A (x B φ)) ↔ ψ))
104, 9syl5bb 181 1 (A B → (x B (x = A φ) ↔ ψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  wrex 2301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553
This theorem is referenced by:  ceqsrexbv  2669  ceqsrex2v  2670  f1oiso  5408  creur  7652  creui  7653
  Copyright terms: Public domain W3C validator