Theorem sbc2iedv 2933

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)

Hypotheses

Ref	Expression
sbc2iedv.1	⊢ 𝐴 ∈ V
sbc2iedv.2	⊢ 𝐵 ∈ V
sbc2iedv.3	⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
sbc2iedv	⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))

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

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem sbc2iedv

Step	Hyp	Ref	Expression
1		sbc2iedv.1	. . 3 ⊢ 𝐴 ∈ V
2	1	a1i 9	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
3		sbc2iedv.2	. . . 4 ⊢ 𝐵 ∈ V
4	3	a1i 9	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ V)
5		sbc2iedv.3	. . . 4 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)))
6	5	impl 375	. . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
7	4, 6	sbcied 2897	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
8	2, 7	sbcied 2897	1 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1299   ∈ wcel 1448  Vcvv 2641  [wsbc 2862

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sbc 2863

This theorem is referenced by:  dfoprab3  6019