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Theorem gencbval 2619
Description: Change of bound variable using implicit substitution. (Contributed by NM, 17-May-1996.) (Proof rewritten by Jim Kingdon, 20-Jun-2018.)
Hypotheses
Ref Expression
gencbval.1 𝐴 ∈ V
gencbval.2 (𝐴 = 𝑦 → (𝜑𝜓))
gencbval.3 (𝐴 = 𝑦 → (𝜒𝜃))
gencbval.4 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
Assertion
Ref Expression
gencbval (∀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 gencbval
StepHypRef Expression
1 alcom 1383 . 2 (∀𝑥𝑦(𝑦 = 𝐴 → (𝜃𝜓)) ↔ ∀𝑦𝑥(𝑦 = 𝐴 → (𝜃𝜓)))
2 gencbval.1 . . . 4 𝐴 ∈ V
3 gencbval.3 . . . . . . 7 (𝐴 = 𝑦 → (𝜒𝜃))
4 gencbval.2 . . . . . . 7 (𝐴 = 𝑦 → (𝜑𝜓))
53, 4imbi12d 227 . . . . . 6 (𝐴 = 𝑦 → ((𝜒𝜑) ↔ (𝜃𝜓)))
65bicomd 133 . . . . 5 (𝐴 = 𝑦 → ((𝜃𝜓) ↔ (𝜒𝜑)))
76eqcoms 2059 . . . 4 (𝑦 = 𝐴 → ((𝜃𝜓) ↔ (𝜒𝜑)))
82, 7ceqsalv 2601 . . 3 (∀𝑦(𝑦 = 𝐴 → (𝜃𝜓)) ↔ (𝜒𝜑))
98albii 1375 . 2 (∀𝑥𝑦(𝑦 = 𝐴 → (𝜃𝜓)) ↔ ∀𝑥(𝜒𝜑))
10 19.23v 1779 . . . 4 (∀𝑥(𝑦 = 𝐴 → (𝜃𝜓)) ↔ (∃𝑥 𝑦 = 𝐴 → (𝜃𝜓)))
11 gencbval.4 . . . . . . 7 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
12 eqcom 2058 . . . . . . . . . 10 (𝐴 = 𝑦𝑦 = 𝐴)
1312biimpi 117 . . . . . . . . 9 (𝐴 = 𝑦𝑦 = 𝐴)
1413adantl 266 . . . . . . . 8 ((𝜒𝐴 = 𝑦) → 𝑦 = 𝐴)
1514eximi 1507 . . . . . . 7 (∃𝑥(𝜒𝐴 = 𝑦) → ∃𝑥 𝑦 = 𝐴)
1611, 15sylbi 118 . . . . . 6 (𝜃 → ∃𝑥 𝑦 = 𝐴)
17 pm2.04 80 . . . . . 6 ((∃𝑥 𝑦 = 𝐴 → (𝜃𝜓)) → (𝜃 → (∃𝑥 𝑦 = 𝐴𝜓)))
1816, 17mpdi 42 . . . . 5 ((∃𝑥 𝑦 = 𝐴 → (𝜃𝜓)) → (𝜃𝜓))
19 ax-1 5 . . . . 5 ((𝜃𝜓) → (∃𝑥 𝑦 = 𝐴 → (𝜃𝜓)))
2018, 19impbii 121 . . . 4 ((∃𝑥 𝑦 = 𝐴 → (𝜃𝜓)) ↔ (𝜃𝜓))
2110, 20bitri 177 . . 3 (∀𝑥(𝑦 = 𝐴 → (𝜃𝜓)) ↔ (𝜃𝜓))
2221albii 1375 . 2 (∀𝑦𝑥(𝑦 = 𝐴 → (𝜃𝜓)) ↔ ∀𝑦(𝜃𝜓))
231, 9, 223bitr3i 203 1 (∀𝑥(𝜒𝜑) ↔ ∀𝑦(𝜃𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by: (None)
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