Theorem sbccsb 4436

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 2718	. . 3 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
2	1	sbcbii 3846	. 2 ⊢ ([𝐴 / 𝑥]𝑦 ∈ {𝑦 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)
3		sbcel2 4418	. 2 ⊢ ([𝐴 / 𝑥]𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})
4	2, 3	bitr3i 277	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝑦 ∈ ⦋𝐴 / 𝑥⦌{𝑦 ∣ 𝜑})

Syntax hints:   ↔ wb 206   ∈ wcel 2108  {cab 2714  [wsbc 3788  ⦋csb 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-nul 4334

This theorem is referenced by: (None)