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

Theorem eqerlem 6559
Description: Lemma for eqer 6560. (Contributed by NM, 17-Mar-2008.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)
Hypotheses
Ref Expression
eqer.1 (𝑥 = 𝑦𝐴 = 𝐵)
eqer.2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
Assertion
Ref Expression
eqerlem (𝑧𝑅𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
Distinct variable groups:   𝑥,𝑤,𝑦   𝑥,𝑧,𝑦   𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥,𝑧,𝑤)   𝐵(𝑦,𝑧,𝑤)   𝑅(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem eqerlem
StepHypRef Expression
1 eqer.2 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝐴 = 𝐵}
21brabsb 4257 . 2 (𝑧𝑅𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵)
3 vex 2740 . . 3 𝑧 ∈ V
4 nfcsb1v 3090 . . . . 5 𝑥𝑧 / 𝑥𝐴
5 nfcsb1v 3090 . . . . 5 𝑥𝑤 / 𝑥𝐴
64, 5nfeq 2327 . . . 4 𝑥𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴
7 vex 2740 . . . . . 6 𝑤 ∈ V
8 nfv 1528 . . . . . . 7 𝑦 𝐴 = 𝑤 / 𝑥𝐴
9 vex 2740 . . . . . . . . . 10 𝑦 ∈ V
10 nfcv 2319 . . . . . . . . . 10 𝑥𝐵
11 eqer.1 . . . . . . . . . 10 (𝑥 = 𝑦𝐴 = 𝐵)
129, 10, 11csbief 3101 . . . . . . . . 9 𝑦 / 𝑥𝐴 = 𝐵
13 csbeq1 3060 . . . . . . . . 9 (𝑦 = 𝑤𝑦 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
1412, 13eqtr3id 2224 . . . . . . . 8 (𝑦 = 𝑤𝐵 = 𝑤 / 𝑥𝐴)
1514eqeq2d 2189 . . . . . . 7 (𝑦 = 𝑤 → (𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴))
168, 15sbciegf 2994 . . . . . 6 (𝑤 ∈ V → ([𝑤 / 𝑦]𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴))
177, 16ax-mp 5 . . . . 5 ([𝑤 / 𝑦]𝐴 = 𝐵𝐴 = 𝑤 / 𝑥𝐴)
18 csbeq1a 3066 . . . . . 6 (𝑥 = 𝑧𝐴 = 𝑧 / 𝑥𝐴)
1918eqeq1d 2186 . . . . 5 (𝑥 = 𝑧 → (𝐴 = 𝑤 / 𝑥𝐴𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
2017, 19bitrid 192 . . . 4 (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝐴 = 𝐵𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
216, 20sbciegf 2994 . . 3 (𝑧 ∈ V → ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴))
223, 21ax-mp 5 . 2 ([𝑧 / 𝑥][𝑤 / 𝑦]𝐴 = 𝐵𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
232, 22bitri 184 1 (𝑧𝑅𝑤𝑧 / 𝑥𝐴 = 𝑤 / 𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  Vcvv 2737  [wsbc 2962  csb 3057   class class class wbr 4000  {copab 4060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062
This theorem is referenced by:  eqer  6560
  Copyright terms: Public domain W3C validator