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Theorem rexcom4b 2573
 Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
Hypothesis
Ref Expression
rexcom4b.1 B V
Assertion
Ref Expression
rexcom4b (xy A (φ x = B) ↔ y A φ)
Distinct variable groups:   x,A   x,y   φ,x   x,B
Allowed substitution hints:   φ(y)   A(y)   B(y)

Proof of Theorem rexcom4b
StepHypRef Expression
1 rexcom4a 2572 . 2 (xy A (φ x = B) ↔ y A (φ x x = B))
2 rexcom4b.1 . . . . 5 B V
32isseti 2557 . . . 4 x x = B
43biantru 286 . . 3 (φ ↔ (φ x x = B))
54rexbii 2325 . 2 (y A φy A (φ x x = B))
61, 5bitr4i 176 1 (xy A (φ x = B) ↔ y A φ)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  Vcvv 2551 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: (None)
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