Theorem sbcimg 3799

Description: Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)

Assertion

Ref	Expression
sbcimg	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))

Proof of Theorem sbcimg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3754	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ [𝐴 / 𝑥](𝜑 → 𝜓)))
2		dfsbcq2 3754	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3		dfsbcq2 3754	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓))
4	2, 3	imbi12d 347	. 2 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
5		sbim 2344	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
6	1, 4, 5	vtoclbg 3531	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567  [wsb 2097   ∈ wcel 2149  [wsbc 3751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-sbc 3752

This theorem is referenced by:  sbc19.21g  3822  sbcssg  4485  iota4an  6519  sbcfung  6561  riotass2  7398  tfinds2  7860  telgsums  20063  bnj110  35191  bnj92  35195  bnj539  35224  bnj540  35225  f1omptsnlem  37905  mptsnunlem  37907  topdifinffinlem  37916  relowlpssretop  37933  rdgeqoa  37939  sbcimi  38684  cdlemkid3N  41632  cdlemkid4  41633  cdlemk35s  41636  cdlemk39s  41638  cdlemk42  41640  frege77  44593  frege116  44632  frege118  44634  sbcim2g  45174  onfrALTlem5  45178  sbcim2gVD  45510  sbcssgVD  45518  onfrALTlem5VD  45520  iccelpart  48106