Theorem sbcimg 3790

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 3744	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ [𝐴 / 𝑥](𝜑 → 𝜓)))
2		dfsbcq2 3744	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3		dfsbcq2 3744	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓))
4	2, 3	imbi12d 344	. 2 ⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))
5		sbim 2310	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
6	1, 4, 5	vtoclbg 3515	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  [wsb 2068   ∈ wcel 2114  [wsbc 3741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-sbc 3742

This theorem is referenced by:  sbc19.21g  3813  sbcssg  4475  iota4an  6475  sbcfung  6517  riotass2  7347  tfinds2  7808  telgsums  19926  bnj110  34995  bnj92  34999  bnj539  35028  bnj540  35029  f1omptsnlem  37512  mptsnunlem  37514  topdifinffinlem  37523  relowlpssretop  37540  rdgeqoa  37546  sbcimi  38282  cdlemkid3N  41230  cdlemkid4  41231  cdlemk35s  41234  cdlemk39s  41236  cdlemk42  41238  frege77  44217  frege116  44256  frege118  44258  sbcim2g  44815  onfrALTlem5  44819  sbcim2gVD  45151  sbcssgVD  45159  onfrALTlem5VD  45161  iccelpart  47715