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Theorem sbcbi2 3834
Description: Substituting into equivalent wff's gives equivalent results. (Contributed by Giovanni Mascellani, 9-Apr-2018.) (Proof shortened by Wolf Lammen, 4-May-2023.) Avoid ax-10, ax-12. (Revised by Steven Nguyen, 5-May-2024.)
Assertion
Ref Expression
sbcbi2 (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))

Proof of Theorem sbcbi2
StepHypRef Expression
1 abbi 2794 . . 3 (∀𝑥(𝜑𝜓) → {𝑥𝜑} = {𝑥𝜓})
2 eleq2 2816 . . 3 ({𝑥𝜑} = {𝑥𝜓} → (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜓}))
31, 2syl 17 . 2 (∀𝑥(𝜑𝜓) → (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜓}))
4 df-sbc 3773 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
5 df-sbc 3773 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
63, 4, 53bitr4g 314 1 (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531   = wceq 1533  wcel 2098  {cab 2703  [wsbc 3772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-sbc 3773
This theorem is referenced by:  csbeq2  3893
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