Theorem dfsbcq2 2992

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 1777 and substitution for class variables df-sbc 2990. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 2991. (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 2259	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
2		df-clab 2183	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
3		df-sbc 2990	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
4	3	bicomi 132	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)
5	1, 2, 4	3bitr3g 222	1 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  [wsb 1776   ∈ wcel 2167  {cab 2182  [wsbc 2989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-clab 2183  df-cleq 2189  df-clel 2192  df-sbc 2990

This theorem is referenced by:  sbsbc  2993  sbc8g  2997  sbceq1a  2999  sbc5  3013  sbcng  3030  sbcimg  3031  sbcan  3032  sbcang  3033  sbcor  3034  sbcorg  3035  sbcbig  3036  sbcal  3041  sbcalg  3042  sbcex2  3043  sbcexg  3044  sbcel1v  3052  sbctt  3056  sbcralt  3066  sbcrext  3067  sbcralg  3068  sbcreug  3070  rspsbc  3072  rspesbca  3074  sbcel12g  3099  sbceqg  3100  sbcbrg  4087  csbopabg  4111  opelopabsb  4294  findes  4639  iota4  5238  csbiotag  5251  csbriotag  5890  nn0ind-raph  9443  uzind4s  9664  bezoutlemmain  12165  bezoutlemex  12168