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Theorem bnj610 31155
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 3354 . . . 4 𝑦 ∈ V
2 bnj610.3 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓′))
31, 2sbcie 3622 . . 3 ([𝑦 / 𝑥]𝜑𝜓′)
43sbcbii 3643 . 2 ([𝐴 / 𝑦][𝑦 / 𝑥]𝜑[𝐴 / 𝑦]𝜓′)
5 sbcco 3610 . 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 3622 . 2 ([𝐴 / 𝑦]𝜓′𝜓)
94, 5, 83bitr3i 290 1 ([𝐴 / 𝑥]𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1631  wcel 2145  Vcvv 3351  [wsbc 3587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588
This theorem is referenced by:  bnj611  31326  bnj1000  31349
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