ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  isarep1 Structured version   GIF version

Theorem isarep1 4900
Description: Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by φ(x, y) i.e. the class ({⟨x, y⟩ ∣ φ} “ A). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
isarep1 (𝑏 ({⟨x, y⟩ ∣ φ} “ A) ↔ x A [𝑏 / y]φ)
Distinct variable groups:   x,A   x,𝑏,y
Allowed substitution hints:   φ(x,y,𝑏)   A(y,𝑏)

Proof of Theorem isarep1
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 vex 2530 . . 3 𝑏 V
21elima 4589 . 2 (𝑏 ({⟨x, y⟩ ∣ φ} “ A) ↔ z A z{⟨x, y⟩ ∣ φ}𝑏)
3 df-br 3729 . . . 4 (z{⟨x, y⟩ ∣ φ}𝑏 ↔ ⟨z, 𝑏 {⟨x, y⟩ ∣ φ})
4 opelopabsb 3961 . . . 4 (⟨z, 𝑏 {⟨x, y⟩ ∣ φ} ↔ [z / x][𝑏 / y]φ)
5 sbsbc 2737 . . . . . 6 ([𝑏 / y]φ[𝑏 / y]φ)
65sbbii 1622 . . . . 5 ([z / x][𝑏 / y]φ ↔ [z / x][𝑏 / y]φ)
7 sbsbc 2737 . . . . 5 ([z / x][𝑏 / y]φ[z / x][𝑏 / y]φ)
86, 7bitr2i 174 . . . 4 ([z / x][𝑏 / y]φ ↔ [z / x][𝑏 / y]φ)
93, 4, 83bitri 195 . . 3 (z{⟨x, y⟩ ∣ φ}𝑏 ↔ [z / x][𝑏 / y]φ)
109rexbii 2301 . 2 (z A z{⟨x, y⟩ ∣ φ}𝑏z A [z / x][𝑏 / y]φ)
11 nfs1v 1789 . . 3 x[z / x][𝑏 / y]φ
12 nfv 1395 . . 3 z[𝑏 / y]φ
13 sbequ12r 1629 . . 3 (z = x → ([z / x][𝑏 / y]φ ↔ [𝑏 / y]φ))
1411, 12, 13cbvrex 2500 . 2 (z A [z / x][𝑏 / y]φx A [𝑏 / y]φ)
152, 10, 143bitri 195 1 (𝑏 ({⟨x, y⟩ ∣ φ} “ A) ↔ x A [𝑏 / y]φ)
Colors of variables: wff set class
Syntax hints:  wb 98   wcel 1367  [wsb 1619  wrex 2277  [wsbc 2733  cop 3343   class class class wbr 3728  {copab 3781  cima 4264
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-v 2529  df-sbc 2734  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-br 3729  df-opab 3783  df-xp 4267  df-cnv 4269  df-dm 4271  df-rn 4272  df-res 4273  df-ima 4274
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator