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Theorem rexcom4b 3225
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 3224 . 2 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
2 rexcom4b.1 . . . . 5 𝐵 ∈ V
32isseti 3207 . . . 4 𝑥 𝑥 = 𝐵
43biantru 526 . . 3 (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
54rexbii 3039 . 2 (∃𝑦𝐴 𝜑 ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
61, 5bitr4i 267 1 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1482  wex 1703  wcel 1989  wrex 2912  Vcvv 3198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-v 3200
This theorem is referenced by:  islshpat  34130
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