Theorem sbc2iedf 33062

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 486	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑊)
4		sbc2iedf.7	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
5	4	anassrs 473	. . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝐴) ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒))
6		sbc2iedf.2	. . . 4 ⊢ Ⅎ𝑦𝜑
7		nfv 1946	. . . 4 ⊢ Ⅎ𝑦 𝑥 = 𝐴
8	6, 7	nfan 1931	. . 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 3780	. 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 3780	1 ⊢ (𝜑 → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜓 ↔ 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  Ⅎwnf 1815   ∈ wcel 2144  [wsbc 3738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-sbc 3739

This theorem is used by:  rspc2daf  33063