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Theorem sbcbi2 2954
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 2251 . . 3 (∀𝑥(𝜑𝜓) ↔ {𝑥𝜑} = {𝑥𝜓})
2 eleq2 2201 . . 3 ({𝑥𝜑} = {𝑥𝜓} → (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜓}))
31, 2sylbi 120 . 2 (∀𝑥(𝜑𝜓) → (𝐴 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜓}))
4 df-sbc 2905 . 2 ([𝐴 / 𝑥]𝜑𝐴 ∈ {𝑥𝜑})
5 df-sbc 2905 . 2 ([𝐴 / 𝑥]𝜓𝐴 ∈ {𝑥𝜓})
63, 4, 53bitr4g 222 1 (∀𝑥(𝜑𝜓) → ([𝐴 / 𝑥]𝜑[𝐴 / 𝑥]𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1329   = wceq 1331  wcel 1480  {cab 2123  [wsbc 2904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-sbc 2905
This theorem is referenced by:  csbeq2  3021
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