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Theorem bj-rexvw 34336
 Description: A weak version of rexv 3467 not using ax-ext 2770 (nor df-cleq 2791, df-clel 2870, df-v 3443), and only core FOL axioms. See also bj-ralvw 34335. (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 3112 . 2 (∃𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝑦𝜓} ∧ 𝜑))
2 bj-rexvw.1 . . . . 5 𝜓
32vexw 2782 . . . 4 𝑥 ∈ {𝑦𝜓}
43biantrur 534 . . 3 (𝜑 ↔ (𝑥 ∈ {𝑦𝜓} ∧ 𝜑))
54exbii 1849 . 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 1781   ∈ wcel 2111  {cab 2776  ∃wrex 3107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-rex 3112 This theorem is referenced by: (None)
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