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

Theorem cbvreu 3410
Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker cbvreuw 3393 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvrmo.1 𝑦𝜑
cbvrmo.2 𝑥𝜓
cbvrmo.3 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvreu (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbvreu
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1909 . . . 4 𝑧(𝑥𝐴𝜑)
21sb8eu 2588 . . 3 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑧[𝑧 / 𝑥](𝑥𝐴𝜑))
3 sban 2075 . . . 4 ([𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
43eubii 2573 . . 3 (∃!𝑧[𝑧 / 𝑥](𝑥𝐴𝜑) ↔ ∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑))
5 clelsb1 2852 . . . . . 6 ([𝑧 / 𝑥]𝑥𝐴𝑧𝐴)
65anbi1i 622 . . . . 5 (([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
76eubii 2573 . . . 4 (∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑))
8 nfv 1909 . . . . . 6 𝑦 𝑧𝐴
9 cbvrmo.1 . . . . . . 7 𝑦𝜑
109nfsb 2516 . . . . . 6 𝑦[𝑧 / 𝑥]𝜑
118, 10nfan 1894 . . . . 5 𝑦(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑)
12 nfv 1909 . . . . 5 𝑧(𝑦𝐴𝜓)
13 eleq1w 2808 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
14 sbequ 2078 . . . . . . 7 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
15 cbvrmo.2 . . . . . . . 8 𝑥𝜓
16 cbvrmo.3 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜓))
1715, 16sbie 2495 . . . . . . 7 ([𝑦 / 𝑥]𝜑𝜓)
1814, 17bitrdi 286 . . . . . 6 (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑𝜓))
1913, 18anbi12d 630 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑦𝐴𝜓)))
2011, 12, 19cbveu 2595 . . . 4 (∃!𝑧(𝑧𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
217, 20bitri 274 . . 3 (∃!𝑧([𝑧 / 𝑥]𝑥𝐴 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
222, 4, 213bitri 296 . 2 (∃!𝑥(𝑥𝐴𝜑) ↔ ∃!𝑦(𝑦𝐴𝜓))
23 df-reu 3364 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑥(𝑥𝐴𝜑))
24 df-reu 3364 . 2 (∃!𝑦𝐴 𝜓 ↔ ∃!𝑦(𝑦𝐴𝜓))
2522, 23, 243bitr4i 302 1 (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wnf 1777  [wsb 2059  wcel 2098  ∃!weu 2556  ∃!wreu 3361
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-10 2129  ax-11 2146  ax-12 2166  ax-13 2365
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clel 2802  df-reu 3364
This theorem is referenced by:  cbvrmo  3411  cbvreuv  3413
  Copyright terms: Public domain W3C validator