Theorem sbcied 3068

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)

Hypotheses

Ref	Expression
sbcied.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
sbcied.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
sbcied	⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem sbcied

Step	Hyp	Ref	Expression
1		sbcied.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		sbcied.2	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
3		nfv 1576	. 2 ⊢ Ⅎ𝑥𝜑
4		nfvd 1577	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
5	1, 2, 3, 4	sbciedf 3067	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  [wsbc 3031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032

This theorem is referenced by:  sbcied2  3069  sbc2iedv  3104  sbc3ie  3105  sbcralt  3108  sbcrext  3109  euotd  4347  riota5f  5998  wrdind  11307  issrg  13984  islmod  14311