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Theorem bnj610 34222
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 3477 . . . 4 𝑦 ∈ V
2 bnj610.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓′))
31, 2sbcie 3820 . . 3 ([𝑦 / 𝑥]𝜑𝜓′)
43sbcbii 3837 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑦]𝜓′)
5 sbccow 3800 . 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 3820 . 2 ([𝐴 / 𝑦]𝜓′𝜓)
94, 5, 83bitr3i 301 1 ([𝐴 / 𝑥]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wcel 2105  Vcvv 3473  [wsbc 3777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-sbc 3778
This theorem is referenced by:  bnj611  34393  bnj1000  34416
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