Theorem sbccsb 4387

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 2715	. . 3 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
2	1	sbcbii 3795	. 2 ⊢ ([𝐴 / 𝑥]𝑦 ∈ {𝑦 ∣ 𝜑} ↔ [𝐴 / 𝑥]𝜑)
3		sbcel2 4369	. 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 2113  {cab 2711  [wsbc 3738  ⦋csb 3847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-nul 4285

This theorem is referenced by: (None)