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 2100 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 2850	. 2 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
2		df-clab 2741	. 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 2099   ∈ wcel 2145  {cab 2740  [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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clab 2741  df-cleq 2754  df-clel 2837  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  3822  sbcreu  3826  rspsbc  3829  rspesbca  3831  sbcel12  4372  sbceqg  4373  csbif  4543  rexreusng  4643  sbcbr123  5163  opelopabsb  5512  csbopab  5538  csbopabw  5539  iota4  6518  csbiota  6530  csbriota  7389  onminex  7805  findes  7901  nn0ind-raph  12725  uzind4s  12961  nn0min  33299