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

Theorem bj-rcleqf 34330
Description: Relative version of cleqf 3008. (Contributed by BJ, 27-Dec-2023.)
Hypotheses
Ref Expression
bj-rcleqf.a 𝑥𝐴
bj-rcleqf.b 𝑥𝐵
bj-rcleqf.v 𝑥𝑉
Assertion
Ref Expression
bj-rcleqf ((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))

Proof of Theorem bj-rcleqf
StepHypRef Expression
1 elin 4167 . . . . 5 (𝑥 ∈ (𝑉𝐴) ↔ (𝑥𝑉𝑥𝐴))
2 elin 4167 . . . . 5 (𝑥 ∈ (𝑉𝐵) ↔ (𝑥𝑉𝑥𝐵))
31, 2bibi12i 342 . . . 4 ((𝑥 ∈ (𝑉𝐴) ↔ 𝑥 ∈ (𝑉𝐵)) ↔ ((𝑥𝑉𝑥𝐴) ↔ (𝑥𝑉𝑥𝐵)))
4 pm5.32 576 . . . 4 ((𝑥𝑉 → (𝑥𝐴𝑥𝐵)) ↔ ((𝑥𝑉𝑥𝐴) ↔ (𝑥𝑉𝑥𝐵)))
53, 4bitr4i 280 . . 3 ((𝑥 ∈ (𝑉𝐴) ↔ 𝑥 ∈ (𝑉𝐵)) ↔ (𝑥𝑉 → (𝑥𝐴𝑥𝐵)))
65albii 1813 . 2 (∀𝑥(𝑥 ∈ (𝑉𝐴) ↔ 𝑥 ∈ (𝑉𝐵)) ↔ ∀𝑥(𝑥𝑉 → (𝑥𝐴𝑥𝐵)))
7 bj-rcleqf.v . . . 4 𝑥𝑉
8 bj-rcleqf.a . . . 4 𝑥𝐴
97, 8nfin 4191 . . 3 𝑥(𝑉𝐴)
10 bj-rcleqf.b . . . 4 𝑥𝐵
117, 10nfin 4191 . . 3 𝑥(𝑉𝐵)
129, 11cleqf 3008 . 2 ((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥(𝑥 ∈ (𝑉𝐴) ↔ 𝑥 ∈ (𝑉𝐵)))
13 df-ral 3141 . 2 (∀𝑥𝑉 (𝑥𝐴𝑥𝐵) ↔ ∀𝑥(𝑥𝑉 → (𝑥𝐴𝑥𝐵)))
146, 12, 133bitr4i 305 1 ((𝑉𝐴) = (𝑉𝐵) ↔ ∀𝑥𝑉 (𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1528   = wceq 1530  wcel 2107  wnfc 2959  wral 3136  cin 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rab 3145  df-v 3495  df-in 3941
This theorem is referenced by:  bj-rcleq  34331  bj-reabeq  34332
  Copyright terms: Public domain W3C validator