Theorem sbcimg 3789

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 3743	. 2 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ [𝐴 / 𝑥](𝜑 → 𝜓)))
2		dfsbcq2 3743	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3		dfsbcq2 3743	. . 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 2309	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
6	1, 4, 5	vtoclbg 3514	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥](𝜑 → 𝜓) ↔ ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541  [wsb 2067   ∈ wcel 2113  [wsbc 3740

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-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-sbc 3741

This theorem is referenced by:  sbc19.21g  3812  sbcssg  4474  iota4an  6474  sbcfung  6516  riotass2  7345  tfinds2  7806  telgsums  19924  bnj110  35016  bnj92  35020  bnj539  35049  bnj540  35050  f1omptsnlem  37543  mptsnunlem  37545  topdifinffinlem  37554  relowlpssretop  37571  rdgeqoa  37577  sbcimi  38313  cdlemkid3N  41215  cdlemkid4  41216  cdlemk35s  41219  cdlemk39s  41221  cdlemk42  41223  frege77  44202  frege116  44241  frege118  44243  sbcim2g  44800  onfrALTlem5  44804  sbcim2gVD  45136  sbcssgVD  45144  onfrALTlem5VD  45146  iccelpart  47700