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Theorem gencbvex 2735
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 1643 . 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 465 . . . . . 6 (𝐴 = 𝑦 → ((𝜒𝜑) ↔ (𝜃𝜓)))
65bicomd 140 . . . . 5 (𝐴 = 𝑦 → ((𝜃𝜓) ↔ (𝜒𝜑)))
76eqcoms 2143 . . . 4 (𝑦 = 𝐴 → ((𝜃𝜓) ↔ (𝜒𝜑)))
82, 7ceqsexv 2728 . . 3 (∃𝑦(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ (𝜒𝜑))
98exbii 1585 . 2 (∃𝑥𝑦(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ ∃𝑥(𝜒𝜑))
10 19.41v 1875 . . . 4 (∃𝑥(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ (∃𝑥 𝑦 = 𝐴 ∧ (𝜃𝜓)))
11 simpr 109 . . . . 5 ((∃𝑥 𝑦 = 𝐴 ∧ (𝜃𝜓)) → (𝜃𝜓))
12 gencbvex.4 . . . . . . . 8 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
13 eqcom 2142 . . . . . . . . . . 11 (𝐴 = 𝑦𝑦 = 𝐴)
1413biimpi 119 . . . . . . . . . 10 (𝐴 = 𝑦𝑦 = 𝐴)
1514adantl 275 . . . . . . . . 9 ((𝜒𝐴 = 𝑦) → 𝑦 = 𝐴)
1615eximi 1580 . . . . . . . 8 (∃𝑥(𝜒𝐴 = 𝑦) → ∃𝑥 𝑦 = 𝐴)
1712, 16sylbi 120 . . . . . . 7 (𝜃 → ∃𝑥 𝑦 = 𝐴)
1817adantr 274 . . . . . 6 ((𝜃𝜓) → ∃𝑥 𝑦 = 𝐴)
1918ancri 322 . . . . 5 ((𝜃𝜓) → (∃𝑥 𝑦 = 𝐴 ∧ (𝜃𝜓)))
2011, 19impbii 125 . . . 4 ((∃𝑥 𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ (𝜃𝜓))
2110, 20bitri 183 . . 3 (∃𝑥(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ (𝜃𝜓))
2221exbii 1585 . 2 (∃𝑦𝑥(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ ∃𝑦(𝜃𝜓))
231, 9, 223bitr3i 209 1 (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wex 1469  wcel 1481  Vcvv 2689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-v 2691
This theorem is referenced by:  gencbvex2  2736
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