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Theorem bj-ralvw 35697
Description: A weak version of ralv 3499 not using ax-ext 2704 (nor df-cleq 2725, df-clel 2811, df-v 3477), and only core FOL axioms. See also bj-rexvw 35698. The analogues for reuv 3501 and rmov 3502 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 3063 . 2 (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜓} → 𝜑))
2 bj-ralvw.1 . . . . 5 𝜓
32vexw 2716 . . . 4 𝑥 ∈ {𝑦𝜓}
43a1bi 363 . . 3 (𝜑 ↔ (𝑥 ∈ {𝑦𝜓} → 𝜑))
54albii 1822 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜓} → 𝜑))
61, 5bitr4i 278 1 (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1540  wcel 2107  {cab 2710  wral 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812
This theorem depends on definitions:  df-bi 206  df-sb 2069  df-clab 2711  df-ral 3063
This theorem is referenced by: (None)
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