Theorem sbciedf 3772

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 413	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
6	3, 5	alrimi 2225	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
7		sbciegft 3767	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
8	1, 2, 6, 7	syl3anc 1379	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119  [wsbc 3730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1094  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-sbc 3731

This theorem is referenced by:  sbc2iegf  3804  csbiebt  3867  sbcnestgfw  4356  sbcnestgf  4361  ovmpodxf  7513  sbc2iedf  32559  reuf1odnf  47577  ovmpordxf  48837