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Theorem gencbval 2729
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 1454 . 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 233 . . . . . 6 (𝐴 = 𝑦 → ((𝜒𝜑) ↔ (𝜃𝜓)))
65bicomd 140 . . . . 5 (𝐴 = 𝑦 → ((𝜃𝜓) ↔ (𝜒𝜑)))
76eqcoms 2140 . . . 4 (𝑦 = 𝐴 → ((𝜃𝜓) ↔ (𝜒𝜑)))
82, 7ceqsalv 2711 . . 3 (∀𝑦(𝑦 = 𝐴 → (𝜃𝜓)) ↔ (𝜒𝜑))
98albii 1446 . 2 (∀𝑥𝑦(𝑦 = 𝐴 → (𝜃𝜓)) ↔ ∀𝑥(𝜒𝜑))
10 19.23v 1855 . . . 4 (∀𝑥(𝑦 = 𝐴 → (𝜃𝜓)) ↔ (∃𝑥 𝑦 = 𝐴 → (𝜃𝜓)))
11 gencbval.4 . . . . . . 7 (𝜃 ↔ ∃𝑥(𝜒𝐴 = 𝑦))
12 eqcom 2139 . . . . . . . . . 10 (𝐴 = 𝑦𝑦 = 𝐴)
1312biimpi 119 . . . . . . . . 9 (𝐴 = 𝑦𝑦 = 𝐴)
1413adantl 275 . . . . . . . 8 ((𝜒𝐴 = 𝑦) → 𝑦 = 𝐴)
1514eximi 1579 . . . . . . 7 (∃𝑥(𝜒𝐴 = 𝑦) → ∃𝑥 𝑦 = 𝐴)
1611, 15sylbi 120 . . . . . 6 (𝜃 → ∃𝑥 𝑦 = 𝐴)
17 pm2.04 82 . . . . . 6 ((∃𝑥 𝑦 = 𝐴 → (𝜃𝜓)) → (𝜃 → (∃𝑥 𝑦 = 𝐴𝜓)))
1816, 17mpdi 43 . . . . 5 ((∃𝑥 𝑦 = 𝐴 → (𝜃𝜓)) → (𝜃𝜓))
19 ax-1 6 . . . . 5 ((𝜃𝜓) → (∃𝑥 𝑦 = 𝐴 → (𝜃𝜓)))
2018, 19impbii 125 . . . 4 ((∃𝑥 𝑦 = 𝐴 → (𝜃𝜓)) ↔ (𝜃𝜓))
2110, 20bitri 183 . . 3 (∀𝑥(𝑦 = 𝐴 → (𝜃𝜓)) ↔ (𝜃𝜓))
2221albii 1446 . 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  wal 1329   = wceq 1331  wex 1468  wcel 1480  Vcvv 2681
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683
This theorem is referenced by: (None)
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