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Theorem bj-rexvw 34802
Description: A weak version of rexv 3433 not using ax-ext 2708 (nor df-cleq 2729, df-clel 2816, df-v 3410), and only core FOL axioms. See also bj-ralvw 34801. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-rexvw.1 𝜓
Assertion
Ref Expression
bj-rexvw (∃𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∃𝑥𝜑)

Proof of Theorem bj-rexvw
StepHypRef Expression
1 df-rex 3067 . 2 (∃𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜓} ∧ 𝜑))
2 bj-rexvw.1 . . . . 5 𝜓
32vexw 2720 . . . 4 𝑥 ∈ {𝑦𝜓}
43biantrur 534 . . 3 (𝜑 ↔ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑))
54exbii 1855 . 2 (∃𝑥𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜓} ∧ 𝜑))
61, 5bitr4i 281 1 (∃𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wex 1787  wcel 2110  {cab 2714  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071  df-clab 2715  df-rex 3067
This theorem is referenced by: (None)
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