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Theorem bj-ralvw 32533
Description: A weak version of ralv 3205 not using ax-ext 2601 (nor df-cleq 2614, df-clel 2617, df-v 3188), but using ax-13 2245. For the sake of illustration, the next theorem bj-rexvwv 32534, a weak version of rexv 3206, has a dv condition and avoids dependency on ax-13 2245, while the analogues for reuv 3207 and rmov 3208 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 2912 . 2 (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜓} → 𝜑))
2 bj-ralvw.1 . . . . 5 𝜓
32bj-vexw 32523 . . . 4 𝑥 ∈ {𝑦𝜓}
43a1bi 352 . . 3 (𝜑 ↔ (𝑥 ∈ {𝑦𝜓} → 𝜑))
54albii 1744 . 2 (∀𝑥𝜑 ↔ ∀𝑥(𝑥 ∈ {𝑦𝜓} → 𝜑))
61, 5bitr4i 267 1 (∀𝑥 ∈ {𝑦𝜓}𝜑 ↔ ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wcel 1987  {cab 2607  wral 2907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-sb 1878  df-clab 2608  df-ral 2912
This theorem is referenced by: (None)
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