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

Theorem rusgrnumwwlkslem 29490
Description: Lemma for rusgrnumwwlks 29495. (Contributed by Alexander van der Vekens, 23-Aug-2018.)
Assertion
Ref Expression
rusgrnumwwlkslem (𝑌 ∈ {𝑤𝑍 ∣ (𝑤‘0) = 𝑃} → {𝑤𝑋 ∣ (𝜑𝜓)} = {𝑤𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃𝜓)})
Distinct variable groups:   𝑤,𝑃   𝑤,𝑌   𝑤,𝑍
Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑤)   𝑋(𝑤)

Proof of Theorem rusgrnumwwlkslem
StepHypRef Expression
1 fveq1 6889 . . . 4 (𝑤 = 𝑌 → (𝑤‘0) = (𝑌‘0))
21eqeq1d 2732 . . 3 (𝑤 = 𝑌 → ((𝑤‘0) = 𝑃 ↔ (𝑌‘0) = 𝑃))
32elrab 3682 . 2 (𝑌 ∈ {𝑤𝑍 ∣ (𝑤‘0) = 𝑃} ↔ (𝑌𝑍 ∧ (𝑌‘0) = 𝑃))
4 ibar 527 . . . . 5 ((𝑌‘0) = 𝑃 → ((𝜑𝜓) ↔ ((𝑌‘0) = 𝑃 ∧ (𝜑𝜓))))
5 3anass 1093 . . . . . 6 (((𝑌‘0) = 𝑃𝜑𝜓) ↔ ((𝑌‘0) = 𝑃 ∧ (𝜑𝜓)))
6 3ancoma 1096 . . . . . 6 (((𝑌‘0) = 𝑃𝜑𝜓) ↔ (𝜑 ∧ (𝑌‘0) = 𝑃𝜓))
75, 6bitr3i 276 . . . . 5 (((𝑌‘0) = 𝑃 ∧ (𝜑𝜓)) ↔ (𝜑 ∧ (𝑌‘0) = 𝑃𝜓))
84, 7bitrdi 286 . . . 4 ((𝑌‘0) = 𝑃 → ((𝜑𝜓) ↔ (𝜑 ∧ (𝑌‘0) = 𝑃𝜓)))
98ad2antlr 723 . . 3 (((𝑌𝑍 ∧ (𝑌‘0) = 𝑃) ∧ 𝑤𝑋) → ((𝜑𝜓) ↔ (𝜑 ∧ (𝑌‘0) = 𝑃𝜓)))
109rabbidva 3437 . 2 ((𝑌𝑍 ∧ (𝑌‘0) = 𝑃) → {𝑤𝑋 ∣ (𝜑𝜓)} = {𝑤𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃𝜓)})
113, 10sylbi 216 1 (𝑌 ∈ {𝑤𝑍 ∣ (𝑤‘0) = 𝑃} → {𝑤𝑋 ∣ (𝜑𝜓)} = {𝑤𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃𝜓)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1085   = wceq 1539  wcel 2104  {crab 3430  cfv 6542  0cc0 11112
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1087  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-in 3954  df-ss 3964  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550
This theorem is referenced by:  rusgrnumwwlks  29495
  Copyright terms: Public domain W3C validator