Theorem sbciedf 3623

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 397	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
6	3, 5	alrimi 2238	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒)))
7		sbciegft 3618	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜒 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜓 ↔ 𝜒))) → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))
8	1, 2, 6, 7	syl3anc 1476	1 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382  ∀wal 1629   = wceq 1631  Ⅎwnf 1856   ∈ wcel 2145  [wsbc 3587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-12 2203  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588

This theorem is referenced by:  sbcied  3624  sbc2iegf  3654  csbiebt  3702  sbcnestgf  4140  ovmpt2dxf  6937  ovmpt2rdxf  42640