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Theorem bj-ralvw 34079
Description: A weak version of ralv 3524 not using ax-ext 2797 (nor df-cleq 2818, df-clel 2897, df-v 3501), and only core FOL axioms. See also bj-rexvw 34080. The analogues for reuv 3526 and rmov 3527 are not proved. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-ralvw.1 𝜓
Assertion
Ref Expression
bj-ralvw (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥𝜑)

Proof of Theorem bj-ralvw
StepHypRef Expression
1 df-ral 3147 . 2 (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜓} → 𝜑))
2 bj-ralvw.1 . . . . 5 𝜓
32vexw 2809 . . . 4 𝑥 ∈ {𝑦𝜓}
43a1bi 364 . . 3 (𝜑 ↔ (𝑥 ∈ {𝑦𝜓} → 𝜑))
54albii 1813 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜓} → 𝜑))
61, 5bitr4i 279 1 (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1528  wcel 2107  {cab 2803  wral 3142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803
This theorem depends on definitions:  df-bi 208  df-sb 2063  df-clab 2804  df-ral 3147
This theorem is referenced by: (None)
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