Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ralvw Structured version   Visualization version   GIF version

Theorem bj-ralvw 36063
Description: A weak version of ralv 3498 not using ax-ext 2702 (nor df-cleq 2723, df-clel 2809, df-v 3475), and only core FOL axioms. See also bj-rexvw 36064. The analogues for reuv 3500 and rmov 3501 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 3061 . 2 (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜓} → 𝜑))
2 bj-ralvw.1 . . . . 5 𝜓
32vexw 2714 . . . 4 𝑥 ∈ {𝑦𝜓}
43a1bi 362 . . 3 (𝜑 ↔ (𝑥 ∈ {𝑦𝜓} → 𝜑))
54albii 1820 . 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 1538  wcel 2105  {cab 2708  wral 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810
This theorem depends on definitions:  df-bi 206  df-sb 2067  df-clab 2709  df-ral 3061
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator