Theorem sbccsb 4393

Description: Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbccsb	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑥, 𝑦)

Proof of Theorem sbccsb

Step	Hyp	Ref	Expression
1		abid 2742	. . 3 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
2	1	sbcbii 3794	. 2 ⊢ ([𝐴 / 𝑥]𝑦 ∈ {𝑦 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)
3		sbcel2 4375	. 2 ⊢ ([𝐴 / 𝑥]𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})
4	2, 3	bitr3i 280	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2144  {cab 2738  [wsbc 3738  ⦋csb 3846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 4047  df-nul 4279

This theorem is used by: (None)