Theorem dfsbcq2 3044

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 1812 and substitution for class variables df-sbc 3042. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3043. (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 2295	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
2		df-clab 2219	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
3		df-sbc 3042	. . 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 1398  [wsb 1811   ∈ wcel 2203  {cab 2218  [wsbc 3041

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-clab 2219  df-cleq 2225  df-clel 2228  df-sbc 3042

This theorem is referenced by:  sbsbc  3045  sbc8g  3049  sbceq1a  3051  sbc5  3065  sbcng  3082  sbcimg  3083  sbcan  3084  sbcang  3085  sbcor  3086  sbcorg  3087  sbcbig  3088  sbcal  3093  sbcalg  3094  sbcex2  3095  sbcexg  3096  sbcel1v  3104  sbctt  3108  sbcralt  3118  sbcrext  3119  sbcralg  3120  sbcreug  3122  rspsbc  3125  rspesbca  3127  sbcel12g  3152  sbceqg  3153  sbcbrg  4163  csbopabg  4187  opelopabsb  4377  findes  4724  iota4  5331  csbiotag  5344  csbriotag  6016  nn0ind-raph  9691  uzind4s  9918  bezoutlemmain  12687  bezoutlemex  12690