MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abeq2w Structured version   Visualization version   GIF version

Theorem abeq2w 2815
Description: Version of abeq2 2872 using implicit substitution, which requires fewer axioms. (Contributed by GG and AV, 18-Sep-2024.)
Hypothesis
Ref Expression
abeq2w.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
abeq2w (𝐴 = {𝑥𝜑} ↔ ∀𝑦(𝑦𝐴𝜓))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥)

Proof of Theorem abeq2w
StepHypRef Expression
1 dfcleq 2731 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑦(𝑦𝐴𝑦 ∈ {𝑥𝜑}))
2 df-clab 2716 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
3 abeq2w.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
43sbievw 2095 . . . . 5 ([𝑦 / 𝑥]𝜑𝜓)
52, 4bitri 274 . . . 4 (𝑦 ∈ {𝑥𝜑} ↔ 𝜓)
65bibi2i 338 . . 3 ((𝑦𝐴𝑦 ∈ {𝑥𝜑}) ↔ (𝑦𝐴𝜓))
76albii 1822 . 2 (∀𝑦(𝑦𝐴𝑦 ∈ {𝑥𝜑}) ↔ ∀𝑦(𝑦𝐴𝜓))
81, 7bitri 274 1 (𝐴 = {𝑥𝜑} ↔ ∀𝑦(𝑦𝐴𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537   = wceq 1539  [wsb 2067  wcel 2106  {cab 2715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730
This theorem is referenced by:  vpwex  5300  fineqvpow  33052
  Copyright terms: Public domain W3C validator