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Theorem bj-rexvwv 32060
 Description: A weak version of rexv 3193 not using ax-ext 2590 (nor df-cleq 2603, df-clel 2606, df-v 3175) with an additional dv condition to avoid dependency on ax-13 2234 as well. See bj-ralvw 32059. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-rexvwv.1 𝜓
Assertion
Ref Expression
bj-rexvwv (∃𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∃𝑥𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem bj-rexvwv
StepHypRef Expression
1 df-rex 2902 . 2 (∃𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜓} ∧ 𝜑))
2 bj-rexvwv.1 . . . . 5 𝜓
32bj-vexwv 32051 . . . 4 𝑥 ∈ {𝑦𝜓}
43biantrur 526 . . 3 (𝜑 ↔ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑))
54exbii 1764 . 2 (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜓} ∧ 𝜑))
61, 5bitr4i 266 1 (∃𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897 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-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-sb 1868  df-clab 2597  df-rex 2902 This theorem is referenced by: (None)
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