Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cbvabvw Structured version   Visualization version   GIF version

Theorem cbvabvw 38045
Description: Version of cbvabv 2952 with a distinct variable condition on x and y, which removes dependencies on ax-11 2207 and ax-13 2389. (Contributed by Steven Nguyen, 4-Dec-2022.)
Hypothesis
Ref Expression
cbvabvw.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbvabvw {𝑥𝜑} = {𝑦𝜓}
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvabvw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 equequ1 2129 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
2 cbvabvw.1 . . . . . 6 (𝑥 = 𝑦 → (𝜑𝜓))
31, 2imbi12d 336 . . . . 5 (𝑥 = 𝑦 → ((𝑥 = 𝑧𝜑) ↔ (𝑦 = 𝑧𝜓)))
43cbvalvw 2143 . . . 4 (∀𝑥(𝑥 = 𝑧𝜑) ↔ ∀𝑦(𝑦 = 𝑧𝜓))
5 sb6 2307 . . . 4 ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧𝜑))
6 sb6 2307 . . . 4 ([𝑧 / 𝑦]𝜓 ↔ ∀𝑦(𝑦 = 𝑧𝜓))
74, 5, 63bitr4i 295 . . 3 ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓)
8 df-clab 2812 . . 3 (𝑧 ∈ {𝑥𝜑} ↔ [𝑧 / 𝑥]𝜑)
9 df-clab 2812 . . 3 (𝑧 ∈ {𝑦𝜓} ↔ [𝑧 / 𝑦]𝜓)
107, 8, 93bitr4i 295 . 2 (𝑧 ∈ {𝑥𝜑} ↔ 𝑧 ∈ {𝑦𝜓})
1110eqriv 2822 1 {𝑥𝜑} = {𝑦𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1654   = wceq 1656  [wsb 2067  wcel 2164  {cab 2811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818
This theorem is referenced by:  cbvrabvw  38046
  Copyright terms: Public domain W3C validator