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Theorem rexcom4b 3200
 Description: Specialized existential commutation lemma. (Contributed by Jeff Madsen, 1-Jun-2011.)
Hypothesis
Ref Expression
rexcom4b.1 𝐵 ∈ V
Assertion
Ref Expression
rexcom4b (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 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 3199 . 2 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
2 rexcom4b.1 . . . . 5 𝐵 ∈ V
32isseti 3182 . . . 4 𝑥 𝑥 = 𝐵
43biantru 525 . . 3 (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
54rexbii 3023 . 2 (∃𝑦𝐴 𝜑 ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
61, 5bitr4i 266 1 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175 This theorem is referenced by:  islshpat  33322
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