Theorem dfsbcq2 3745

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 2102 and substitution for class variables df-sbc 3743. Unlike Quine, we use a different syntax for each in order to avoid overloading it. See remarks in dfsbcq 3744. (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 2852	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
2		df-clab 2742	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
3		df-sbc 3743	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
4	3	bicomi 227	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)
5	1, 2, 4	3bitr3g 316	1 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  [wsb 2101   ∈ wcel 2147  {cab 2741  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clab 2742  df-cleq 2755  df-clel 2839  df-sbc 3743

This theorem is used by:  sbsbc  3746  sbc8g  3750  sbc2or  3751  sbceq1a  3753  sbc5ALT  3771  sbcng  3789  sbcimg  3790  sbcan  3791  sbcor  3792  sbcbig  3793  sbcim1  3795  sbcal  3801  sbcex2  3802  sbcel1v  3807  sbctt  3811  sbcralt  3823  sbcreu  3827  rspsbc  3830  rspesbca  3832  sbcel12  4372  sbceqg  4373  csbif  4543  rexreusng  4640  sbcbr123  5164  opelopabsb  5509  csbopab  5535  csbopabw  5536  iota4  6541  csbiota  6554  csbriota  7418  onminex  7843  findes  7939  nn0ind-raph  12829  uzind4s  13065  nn0min  33485