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

Theorem csbopabgALT 5160
Description: Move substitution into a class abstraction. Version of csbopab 5159 with a sethood antecedent but depending on fewer axioms. (Contributed by NM, 6-Aug-2007.) (Proof shortened by Mario Carneiro, 17-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbopabgALT (𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑥)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem csbopabgALT
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3678 . . 3 (𝑤 = 𝐴𝑤 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑})
2 dfsbcq2 3580 . . . 4 (𝑤 = 𝐴 → ([𝑤 / 𝑥]𝜑[𝐴 / 𝑥]𝜑))
32opabbidv 4869 . . 3 (𝑤 = 𝐴 → {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
41, 3eqeq12d 2776 . 2 (𝑤 = 𝐴 → (𝑤 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑} ↔ 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑}))
5 vex 3344 . . 3 𝑤 ∈ V
6 nfs1v 2575 . . . 4 𝑥[𝑤 / 𝑥]𝜑
76nfopab 4871 . . 3 𝑥{⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
8 sbequ12 2259 . . . 4 (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
98opabbidv 4869 . . 3 (𝑥 = 𝑤 → {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑})
105, 7, 9csbief 3700 . 2 𝑤 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝑤 / 𝑥]𝜑}
114, 10vtoclg 3407 1 (𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  [wsb 2047  wcel 2140  [wsbc 3577  csb 3675  {copab 4865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-sbc 3578  df-csb 3676  df-opab 4866
This theorem is referenced by:  csbcnvgALT  5463
  Copyright terms: Public domain W3C validator