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Theorem gencbvex 2656
Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
gencbvex.1 𝐴 ∈ V
gencbvex.2 (𝐴 = 𝑦 → (𝜑𝜓))
gencbvex.3 (𝐴 = 𝑦 → (𝜒𝜃))
gencbvex.4 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
Assertion
Ref Expression
gencbvex (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝜃,𝑥   𝜒,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥)   𝜃(𝑦)   𝐴(𝑥)

Proof of Theorem gencbvex
StepHypRef Expression
1 excom 1595 . 2 (∃𝑥𝑦(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ ∃𝑦𝑥(𝑦 = 𝐴 ∧ (𝜃𝜓)))
2 gencbvex.1 . . . 4 𝐴 ∈ V
3 gencbvex.3 . . . . . . 7 (𝐴 = 𝑦 → (𝜒𝜃))
4 gencbvex.2 . . . . . . 7 (𝐴 = 𝑦 → (𝜑𝜓))
53, 4anbi12d 457 . . . . . 6 (𝐴 = 𝑦 → ((𝜒𝜑) ↔ (𝜃𝜓)))
65bicomd 139 . . . . 5 (𝐴 = 𝑦 → ((𝜃𝜓) ↔ (𝜒𝜑)))
76eqcoms 2086 . . . 4 (𝑦 = 𝐴 → ((𝜃𝜓) ↔ (𝜒𝜑)))
82, 7ceqsexv 2649 . . 3 (∃𝑦(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ (𝜒𝜑))
98exbii 1537 . 2 (∃𝑥𝑦(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ ∃𝑥(𝜒𝜑))
10 19.41v 1825 . . . 4 (∃𝑥(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ (∃𝑥 𝑦 = 𝐴 ∧ (𝜃𝜓)))
11 simpr 108 . . . . 5 ((∃𝑥 𝑦 = 𝐴 ∧ (𝜃𝜓)) → (𝜃𝜓))
12 gencbvex.4 . . . . . . . 8 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
13 eqcom 2085 . . . . . . . . . . 11 (𝐴 = 𝑦𝑦 = 𝐴)
1413biimpi 118 . . . . . . . . . 10 (𝐴 = 𝑦𝑦 = 𝐴)
1514adantl 271 . . . . . . . . 9 ((𝜒𝐴 = 𝑦) → 𝑦 = 𝐴)
1615eximi 1532 . . . . . . . 8 (∃𝑥(𝜒𝐴 = 𝑦) → ∃𝑥 𝑦 = 𝐴)
1712, 16sylbi 119 . . . . . . 7 (𝜃 → ∃𝑥 𝑦 = 𝐴)
1817adantr 270 . . . . . 6 ((𝜃𝜓) → ∃𝑥 𝑦 = 𝐴)
1918ancri 317 . . . . 5 ((𝜃𝜓) → (∃𝑥 𝑦 = 𝐴 ∧ (𝜃𝜓)))
2011, 19impbii 124 . . . 4 ((∃𝑥 𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ (𝜃𝜓))
2110, 20bitri 182 . . 3 (∃𝑥(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ (𝜃𝜓))
2221exbii 1537 . 2 (∃𝑦𝑥(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ ∃𝑦(𝜃𝜓))
231, 9, 223bitr3i 208 1 (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434  Vcvv 2612
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-v 2614
This theorem is referenced by:  gencbvex2  2657
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