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Theorem bj-ralvw 34600
 Description: A weak version of ralv 3435 not using ax-ext 2729 (nor df-cleq 2750, df-clel 2830, df-v 3411), and only core FOL axioms. See also bj-rexvw 34601. The analogues for reuv 3437 and rmov 3438 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 3075 . 2 (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜓} → 𝜑))
2 bj-ralvw.1 . . . . 5 𝜓
32vexw 2741 . . . 4 𝑥 ∈ {𝑦𝜓}
43a1bi 366 . . 3 (𝜑 ↔ (𝑥 ∈ {𝑦𝜓} → 𝜑))
54albii 1821 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜓} → 𝜑))
61, 5bitr4i 281 1 (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2111  {cab 2735  ∀wral 3070 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-sb 2070  df-clab 2736  df-ral 3075 This theorem is referenced by: (None)
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