Theorem sbciedf 3821

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 29-Dec-2014.)

Hypotheses

Ref	Expression
sbcied.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
sbcied.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
sbciedf.3	⊢ Ⅎ𝑥𝜑
sbciedf.4	⊢ (𝜑 → Ⅎ𝑥𝜒)

Assertion

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

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem sbciedf

Step	Hyp	Ref	Expression
1		sbcied.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		sbciedf.4	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
3		sbciedf.3	. . 3 ⊢ Ⅎ𝑥𝜑
4		sbcied.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
5	4	ex 412	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
6	3, 5	alrimi 2205	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
7		sbciegft 3815	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
8	1, 2, 6, 7	syl3anc 1370	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1538   = wceq 1540  Ⅎwnf 1784   ∈ wcel 2105  [wsbc 3777

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-sbc 3778

This theorem is referenced by:  sbciedOLD  3823  sbc2iegf  3859  csbiebt  3923  sbcnestgfw  4418  sbcnestgf  4423  ovmpodxf  7561  sbc2iedf  32141  reuf1odnf  46276  ovmpordxf  47179