Theorem sbc2iedf 30815

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Thierry Arnoux, 4-Jul-2023.)

Hypotheses

Ref	Expression
sbc2iedf.1	⊢ Ⅎ𝑥𝜑
sbc2iedf.2	⊢ Ⅎ𝑦𝜑
sbc2iedf.3	⊢ Ⅎ𝑥𝜒
sbc2iedf.4	⊢ Ⅎ𝑦𝜒
sbc2iedf.5	⊢ (𝜑 → 𝐴 ∈ 𝑉)
sbc2iedf.6	⊢ (𝜑 → 𝐵 ∈ 𝑊)
sbc2iedf.7	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
sbc2iedf	⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem sbc2iedf

Step	Hyp	Ref	Expression
1		sbc2iedf.5	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		sbc2iedf.6	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3	2	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑊)
4		sbc2iedf.7	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
5	4	anassrs 468	. . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
6		sbc2iedf.2	. . . 4 ⊢ Ⅎ𝑦𝜑
7		nfv 1917	. . . 4 ⊢ Ⅎ𝑦 𝑥 = 𝐴
8	6, 7	nfan 1902	. . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 = 𝐴)
9		sbc2iedf.4	. . . 4 ⊢ Ⅎ𝑦𝜒
10	9	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → Ⅎ𝑦𝜒)
11	3, 5, 8, 10	sbciedf 3760	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ([𝐵 / 𝑦]𝜓 ↔ 𝜒))
12		sbc2iedf.1	. 2 ⊢ Ⅎ𝑥𝜑
13		sbc2iedf.3	. . 3 ⊢ Ⅎ𝑥𝜒
14	13	a1i 11	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
15	1, 11, 12, 14	sbciedf 3760	1 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2106  [wsbc 3716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-sbc 3717

This theorem is referenced by:  rspc2daf  30816