Theorem sbc3ie 3877

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
sbc3ie.1	⊢ 𝐴 ∈ V
sbc3ie.2	⊢ 𝐵 ∈ V
sbc3ie.3	⊢ 𝐶 ∈ V
sbc3ie.4	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbc3ie	⊢ ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑 ↔ 𝜓)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑧,𝐶   𝜓,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem sbc3ie

Step	Hyp	Ref	Expression
1		sbc3ie.1	. 2 ⊢ 𝐴 ∈ V
2		sbc3ie.2	. 2 ⊢ 𝐵 ∈ V
3		sbc3ie.3	. . . 4 ⊢ 𝐶 ∈ V
4	3	a1i 11	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 ∈ V)
5		sbc3ie.4	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
6	5	3expa 1117	. . 3 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
7	4, 6	sbcied 3837	. 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ([𝐶 / 𝑧]𝜑 ↔ 𝜓))
8	1, 2, 7	sbc2ie 3874	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦][𝐶 / 𝑧]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  Vcvv 3478  [wsbc 3791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-sbc 3792

This theorem is referenced by:  isdlat  18580  islmod  20879  isslmd  33191  hdmap1fval  41779  hdmapfval  41810  hgmapfval  41869  rmydioph  43003