Theorem dfsbcq2 2912

Description: This theorem, which is similar to Theorem 6.7 of [Quine] p. 42 and holds under both our definition and Quine's, relates logic substitution df-sb 1736 and substitution for class variables df-sbc 2910. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2911. (Contributed by NM, 31-Dec-2016.)

Assertion

Ref	Expression
dfsbcq2	⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))

Proof of Theorem dfsbcq2

Step	Hyp	Ref	Expression
1		eleq1 2202	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
2		df-clab 2126	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
3		df-sbc 2910	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
4	3	bicomi 131	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)
5	1, 2, 4	3bitr3g 221	1 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480  [wsb 1735  {cab 2125  [wsbc 2909

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-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-clab 2126  df-cleq 2132  df-clel 2135  df-sbc 2910

This theorem is referenced by:  sbsbc  2913  sbc8g  2916  sbceq1a  2918  sbc5  2932  sbcng  2949  sbcimg  2950  sbcan  2951  sbcang  2952  sbcor  2953  sbcorg  2954  sbcbig  2955  sbcal  2960  sbcalg  2961  sbcex2  2962  sbcexg  2963  sbcel1v  2971  sbctt  2975  sbcralt  2985  sbcrext  2986  sbcralg  2987  sbcreug  2989  rspsbc  2991  rspesbca  2993  sbcel12g  3017  sbceqg  3018  sbcbrg  3982  csbopabg  4006  opelopabsb  4182  findes  4517  iota4  5106  csbiotag  5116  csbriotag  5742  nn0ind-raph  9168  uzind4s  9385  bezoutlemmain  11686  bezoutlemex  11689