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Theorem bnj610 32013
Description: Pass from equality (𝑥 = 𝐴) to substitution ([𝐴 / 𝑥]) without the distinct variable restriction ($d 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 3498 . . . 4 𝑦 ∈ V
2 bnj610.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓′))
31, 2sbcie 3812 . . 3 ([𝑦 / 𝑥]𝜑𝜓′)
43sbcbii 3829 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑦]𝜓′)
5 sbccow 3795 . 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 3812 . 2 ([𝐴 / 𝑦]𝜓′𝜓)
94, 5, 83bitr3i 303 1 ([𝐴 / 𝑥]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1533  wcel 2110  Vcvv 3495  [wsbc 3772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-12 2172  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3497  df-sbc 3773
This theorem is referenced by:  bnj611  32185  bnj1000  32208
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