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Theorem bnj610 32259
 Description: Pass from equality (𝑥 = 𝐴) to substitution ([𝐴 / 𝑥]) without the distinct variable condition on 𝐴, 𝑥. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj610.1 𝐴 ∈ V
bnj610.2 (𝑥 = 𝐴 → (𝜑𝜓))
bnj610.3 (𝑥 = 𝑦 → (𝜑𝜓′))
bnj610.4 (𝑦 = 𝐴 → (𝜓′𝜓))
Assertion
Ref Expression
bnj610 ([𝐴 / 𝑥]𝜑𝜓)
Distinct variable groups:   𝑦,𝐴   𝜑,𝑦   𝜓,𝑦   𝑥,𝜓′   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)   𝜓′(𝑦)

Proof of Theorem bnj610
StepHypRef Expression
1 vex 3413 . . . 4 𝑦 ∈ V
2 bnj610.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓′))
31, 2sbcie 3739 . . 3 ([𝑦 / 𝑥]𝜑𝜓′)
43sbcbii 3755 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑦]𝜓′)
5 sbccow 3721 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑥]𝜑)
6 bnj610.1 . . 3 𝐴 ∈ V
7 bnj610.4 . . 3 (𝑦 = 𝐴 → (𝜓′𝜓))
86, 7sbcie 3739 . 2 ([𝐴 / 𝑦]𝜓′𝜓)
94, 5, 83bitr3i 304 1 ([𝐴 / 𝑥]𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3409  [wsbc 3698 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-sbc 3699 This theorem is referenced by:  bnj611  32431  bnj1000  32454
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