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Theorem sbcbi2 3647
Description: Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.)
Assertion
Ref Expression
sbcbi2 (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Proof of Theorem sbcbi2
StepHypRef Expression
1 abbi 2880 . . 3 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
2 eleq2 2833 . . 3 ({𝑥𝜑} = {𝑥𝜓} → (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜓}))
31, 2sylbi 208 . 2 (∀𝑥(𝜑𝜓) → (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜓}))
4 df-sbc 3599 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
5 df-sbc 3599 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
63, 4, 53bitr4g 305 1 (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650   = wceq 1652  wcel 2155  {cab 2751  [wsbc 3598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-sbc 3599
This theorem is referenced by:  csbeq2  3697
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