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Theorem gencbvex 3390
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 2191 . 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 749 . . . . . 6 (𝐴 = 𝑦 → ((𝜒𝜑) ↔ (𝜃𝜓)))
65bicomd 213 . . . . 5 (𝐴 = 𝑦 → ((𝜃𝜓) ↔ (𝜒𝜑)))
76eqcoms 2768 . . . 4 (𝑦 = 𝐴 → ((𝜃𝜓) ↔ (𝜒𝜑)))
82, 7ceqsexv 3382 . . 3 (∃𝑦(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ (𝜒𝜑))
98exbii 1923 . 2 (∃𝑥𝑦(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ ∃𝑥(𝜒𝜑))
10 19.41v 2026 . . . 4 (∃𝑥(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ (∃𝑥 𝑦 = 𝐴 ∧ (𝜃𝜓)))
11 simpr 479 . . . . 5 ((∃𝑥 𝑦 = 𝐴 ∧ (𝜃𝜓)) → (𝜃𝜓))
12 gencbvex.4 . . . . . . . 8 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
13 eqcom 2767 . . . . . . . . . . 11 (𝐴 = 𝑦𝑦 = 𝐴)
1413biimpi 206 . . . . . . . . . 10 (𝐴 = 𝑦𝑦 = 𝐴)
1514adantl 473 . . . . . . . . 9 ((𝜒𝐴 = 𝑦) → 𝑦 = 𝐴)
1615eximi 1911 . . . . . . . 8 (∃𝑥(𝜒𝐴 = 𝑦) → ∃𝑥 𝑦 = 𝐴)
1712, 16sylbi 207 . . . . . . 7 (𝜃 → ∃𝑥 𝑦 = 𝐴)
1817adantr 472 . . . . . 6 ((𝜃𝜓) → ∃𝑥 𝑦 = 𝐴)
1918ancri 576 . . . . 5 ((𝜃𝜓) → (∃𝑥 𝑦 = 𝐴 ∧ (𝜃𝜓)))
2011, 19impbii 199 . . . 4 ((∃𝑥 𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ (𝜃𝜓))
2110, 20bitri 264 . . 3 (∃𝑥(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ (𝜃𝜓))
2221exbii 1923 . 2 (∃𝑦𝑥(𝑦 = 𝐴 ∧ (𝜃𝜓)) ↔ ∃𝑦(𝜃𝜓))
231, 9, 223bitr3i 290 1 (∃𝑥(𝜒𝜑) ↔ ∃𝑦(𝜃𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340
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 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-11 2183  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-v 3342
This theorem is referenced by:  gencbvex2  3391  gencbval  3392
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