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

Theorem abbib 2811
Description: Equal class abstractions require equivalent formulas, and conversely. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) Remove dependency on ax-8 2110 and df-clel 2816 (by avoiding use of cleqh 2871). (Revised by BJ, 23-Jun-2019.) Definitial form. (Revised by Wolf Lammen, 23-Feb-2025.)
Assertion
Ref Expression
abbib ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))

Proof of Theorem abbib
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfcleq 2730 . 2 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓}))
2 nfsab1 2722 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
3 nfsab1 2722 . . . 4 𝑥 𝑦 ∈ {𝑥𝜓}
42, 3nfbi 1903 . . 3 𝑥(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓})
5 nfv 1914 . . 3 𝑦(𝜑𝜓)
6 df-clab 2715 . . . . 5 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
7 sbequ12r 2252 . . . . 5 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑𝜑))
86, 7bitrid 283 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
9 df-clab 2715 . . . . 5 (𝑦 ∈ {𝑥𝜓} ↔ [𝑦 / 𝑥]𝜓)
10 sbequ12r 2252 . . . . 5 (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜓𝜓))
119, 10bitrid 283 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜓} ↔ 𝜓))
128, 11bibi12d 345 . . 3 (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓}) ↔ (𝜑𝜓)))
134, 5, 12cbvalv1 2343 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} ↔ 𝑦 ∈ {𝑥𝜓}) ↔ ∀𝑥(𝜑𝜓))
141, 13bitri 275 1 ({𝑥𝜑} = {𝑥𝜓} ↔ ∀𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1538   = wceq 1540  [wsb 2064  wcel 2108  {cab 2714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729
This theorem is referenced by:  eqabb  2881  nabbib  3045  rabbi  3467  ab0  4380  absn  4645  karden  9935  abeqabi  43421  elnev  44457  csbingVD  44904  csbsngVD  44913  csbxpgVD  44914  csbrngVD  44916  csbunigVD  44918  csbfv12gALTVD  44919
  Copyright terms: Public domain W3C validator