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

Theorem cbviota 5756
Description: Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)
Hypotheses
Ref Expression
cbviota.1 (𝑥 = 𝑦 → (𝜑𝜓))
cbviota.2 𝑦𝜑
cbviota.3 𝑥𝜓
Assertion
Ref Expression
cbviota (℩𝑥𝜑) = (℩𝑦𝜓)

Proof of Theorem cbviota
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1829 . . . . . 6 𝑧(𝜑𝑥 = 𝑤)
2 nfs1v 2421 . . . . . . 7 𝑥[𝑧 / 𝑥]𝜑
3 nfv 1829 . . . . . . 7 𝑥 𝑧 = 𝑤
42, 3nfbi 1820 . . . . . 6 𝑥([𝑧 / 𝑥]𝜑𝑧 = 𝑤)
5 sbequ12 2095 . . . . . . 7 (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
6 equequ1 1938 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝑤𝑧 = 𝑤))
75, 6bibi12d 333 . . . . . 6 (𝑥 = 𝑧 → ((𝜑𝑥 = 𝑤) ↔ ([𝑧 / 𝑥]𝜑𝑧 = 𝑤)))
81, 4, 7cbval 2254 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑧([𝑧 / 𝑥]𝜑𝑧 = 𝑤))
9 cbviota.2 . . . . . . . 8 𝑦𝜑
109nfsb 2424 . . . . . . 7 𝑦[𝑧 / 𝑥]𝜑
11 nfv 1829 . . . . . . 7 𝑦 𝑧 = 𝑤
1210, 11nfbi 1820 . . . . . 6 𝑦([𝑧 / 𝑥]𝜑𝑧 = 𝑤)
13 nfv 1829 . . . . . 6 𝑧(𝜓𝑦 = 𝑤)
14 sbequ 2360 . . . . . . . 8 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15 cbviota.3 . . . . . . . . 9 𝑥𝜓
16 cbviota.1 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜓))
1715, 16sbie 2392 . . . . . . . 8 ([𝑦 / 𝑥]𝜑𝜓)
1814, 17syl6bb 274 . . . . . . 7 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
19 equequ1 1938 . . . . . . 7 (𝑧 = 𝑦 → (𝑧 = 𝑤𝑦 = 𝑤))
2018, 19bibi12d 333 . . . . . 6 (𝑧 = 𝑦 → (([𝑧 / 𝑥]𝜑𝑧 = 𝑤) ↔ (𝜓𝑦 = 𝑤)))
2112, 13, 20cbval 2254 . . . . 5 (∀𝑧([𝑧 / 𝑥]𝜑𝑧 = 𝑤) ↔ ∀𝑦(𝜓𝑦 = 𝑤))
228, 21bitri 262 . . . 4 (∀𝑥(𝜑𝑥 = 𝑤) ↔ ∀𝑦(𝜓𝑦 = 𝑤))
2322abbii 2722 . . 3 {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
2423unieqi 4372 . 2 {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)} = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
25 dfiota2 5752 . 2 (℩𝑥𝜑) = {𝑤 ∣ ∀𝑥(𝜑𝑥 = 𝑤)}
26 dfiota2 5752 . 2 (℩𝑦𝜓) = {𝑤 ∣ ∀𝑦(𝜓𝑦 = 𝑤)}
2724, 25, 263eqtr4i 2638 1 (℩𝑥𝜑) = (℩𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472   = wceq 1474  wnf 1698  [wsb 1866  {cab 2592   cuni 4363  cio 5749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-rex 2898  df-sn 4122  df-uni 4364  df-iota 5751
This theorem is referenced by:  cbviotav  5757  fvopab5  6199  cbvriota  6496
  Copyright terms: Public domain W3C validator