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Theorem rexcom4b 2880
 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 2879 . 2 (xy A (φ x = B) ↔ y A (φ x x = B))
2 rexcom4b.1 . . . . 5 B V
32isseti 2865 . . . 4 x x = B
43biantru 491 . . 3 (φ ↔ (φ x x = B))
54rexbii 2639 . 2 (y A φy A (φ x x = B))
61, 5bitr4i 243 1 (xy A (φ x = B) ↔ y A φ)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859 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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861 This theorem is referenced by: (None)
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