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Theorem sseliALT 4719
 Description: Alternate proof of sseli 3564 illustrating the use of the weak deduction theorem to prove it from the inference sselii 3565. (Contributed by NM, 24-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
sseliALT.1 𝐴𝐵
Assertion
Ref Expression
sseliALT (𝐶𝐴𝐶𝐵)

Proof of Theorem sseliALT
StepHypRef Expression
1 biidd 251 . 2 (𝐴 = if(𝐶𝐴, 𝐴, {∅}) → (𝐶𝐵𝐶𝐵))
2 eleq2 2677 . 2 (𝐵 = if(𝐶𝐴, 𝐵, {∅}) → (𝐶𝐵𝐶 ∈ if(𝐶𝐴, 𝐵, {∅})))
3 eleq1 2676 . 2 (𝐶 = if(𝐶𝐴, 𝐶, ∅) → (𝐶 ∈ if(𝐶𝐴, 𝐵, {∅}) ↔ if(𝐶𝐴, 𝐶, ∅) ∈ if(𝐶𝐴, 𝐵, {∅})))
4 sseq1 3589 . . . 4 (𝐴 = if(𝐶𝐴, 𝐴, {∅}) → (𝐴𝐵 ↔ if(𝐶𝐴, 𝐴, {∅}) ⊆ 𝐵))
5 sseq2 3590 . . . 4 (𝐵 = if(𝐶𝐴, 𝐵, {∅}) → (if(𝐶𝐴, 𝐴, {∅}) ⊆ 𝐵 ↔ if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅})))
6 biidd 251 . . . 4 (𝐶 = if(𝐶𝐴, 𝐶, ∅) → (if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅}) ↔ if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅})))
7 sseq1 3589 . . . 4 ({∅} = if(𝐶𝐴, 𝐴, {∅}) → ({∅} ⊆ {∅} ↔ if(𝐶𝐴, 𝐴, {∅}) ⊆ {∅}))
8 sseq2 3590 . . . 4 ({∅} = if(𝐶𝐴, 𝐵, {∅}) → (if(𝐶𝐴, 𝐴, {∅}) ⊆ {∅} ↔ if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅})))
9 biidd 251 . . . 4 (∅ = if(𝐶𝐴, 𝐶, ∅) → (if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅}) ↔ if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅})))
10 sseliALT.1 . . . 4 𝐴𝐵
11 ssid 3587 . . . 4 {∅} ⊆ {∅}
124, 5, 6, 7, 8, 9, 10, 11keephyp3v 4104 . . 3 if(𝐶𝐴, 𝐴, {∅}) ⊆ if(𝐶𝐴, 𝐵, {∅})
13 eleq2 2677 . . . 4 (𝐴 = if(𝐶𝐴, 𝐴, {∅}) → (𝐶𝐴𝐶 ∈ if(𝐶𝐴, 𝐴, {∅})))
14 biidd 251 . . . 4 (𝐵 = if(𝐶𝐴, 𝐵, {∅}) → (𝐶 ∈ if(𝐶𝐴, 𝐴, {∅}) ↔ 𝐶 ∈ if(𝐶𝐴, 𝐴, {∅})))
15 eleq1 2676 . . . 4 (𝐶 = if(𝐶𝐴, 𝐶, ∅) → (𝐶 ∈ if(𝐶𝐴, 𝐴, {∅}) ↔ if(𝐶𝐴, 𝐶, ∅) ∈ if(𝐶𝐴, 𝐴, {∅})))
16 eleq2 2677 . . . 4 ({∅} = if(𝐶𝐴, 𝐴, {∅}) → (∅ ∈ {∅} ↔ ∅ ∈ if(𝐶𝐴, 𝐴, {∅})))
17 biidd 251 . . . 4 ({∅} = if(𝐶𝐴, 𝐵, {∅}) → (∅ ∈ if(𝐶𝐴, 𝐴, {∅}) ↔ ∅ ∈ if(𝐶𝐴, 𝐴, {∅})))
18 eleq1 2676 . . . 4 (∅ = if(𝐶𝐴, 𝐶, ∅) → (∅ ∈ if(𝐶𝐴, 𝐴, {∅}) ↔ if(𝐶𝐴, 𝐶, ∅) ∈ if(𝐶𝐴, 𝐴, {∅})))
19 0ex 4718 . . . . 5 ∅ ∈ V
2019snid 4155 . . . 4 ∅ ∈ {∅}
2113, 14, 15, 16, 17, 18, 20elimhyp3v 4098 . . 3 if(𝐶𝐴, 𝐶, ∅) ∈ if(𝐶𝐴, 𝐴, {∅})
2212, 21sselii 3565 . 2 if(𝐶𝐴, 𝐶, ∅) ∈ if(𝐶𝐴, 𝐵, {∅})
231, 2, 3, 22dedth3v 4094 1 (𝐶𝐴𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126 This theorem is referenced by: (None)
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