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Theorem wfisgOLD 6354
Description: Obsolete version of wfisg 6353 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 11-Feb-2011.)
Hypothesis
Ref Expression
wfisgOLD.1 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
Assertion
Ref Expression
wfisgOLD ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝑦,𝑅,𝑧
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem wfisgOLD
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ssrab2 4076 . . 3 {𝑦𝐴𝜑} ⊆ 𝐴
2 dfss3 3969 . . . . . . . 8 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦𝐴𝜑})
3 nfcv 2901 . . . . . . . . . . 11 𝑦𝐴
43elrabsf 3824 . . . . . . . . . 10 (𝑧 ∈ {𝑦𝐴𝜑} ↔ (𝑧𝐴[𝑧 / 𝑦]𝜑))
54simprbi 495 . . . . . . . . 9 (𝑧 ∈ {𝑦𝐴𝜑} → [𝑧 / 𝑦]𝜑)
65ralimi 3081 . . . . . . . 8 (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)𝑧 ∈ {𝑦𝐴𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
72, 6sylbi 216 . . . . . . 7 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑)
8 nfv 1915 . . . . . . . . 9 𝑦 𝑤𝐴
9 nfcv 2901 . . . . . . . . . . 11 𝑦Pred(𝑅, 𝐴, 𝑤)
10 nfsbc1v 3796 . . . . . . . . . . 11 𝑦[𝑧 / 𝑦]𝜑
119, 10nfralw 3306 . . . . . . . . . 10 𝑦𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑
12 nfsbc1v 3796 . . . . . . . . . 10 𝑦[𝑤 / 𝑦]𝜑
1311, 12nfim 1897 . . . . . . . . 9 𝑦(∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑)
148, 13nfim 1897 . . . . . . . 8 𝑦(𝑤𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))
15 eleq1w 2814 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑦𝐴𝑤𝐴))
16 predeq3 6303 . . . . . . . . . . 11 (𝑦 = 𝑤 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑤))
1716raleqdv 3323 . . . . . . . . . 10 (𝑦 = 𝑤 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑))
18 sbceq1a 3787 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝜑[𝑤 / 𝑦]𝜑))
1917, 18imbi12d 343 . . . . . . . . 9 (𝑦 = 𝑤 → ((∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑) ↔ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑)))
2015, 19imbi12d 343 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑)) ↔ (𝑤𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))))
21 wfisgOLD.1 . . . . . . . 8 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
2214, 20, 21chvarfv 2231 . . . . . . 7 (𝑤𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑤)[𝑧 / 𝑦]𝜑[𝑤 / 𝑦]𝜑))
237, 22syl5 34 . . . . . 6 (𝑤𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → [𝑤 / 𝑦]𝜑))
2423anc2li 554 . . . . 5 (𝑤𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → (𝑤𝐴[𝑤 / 𝑦]𝜑)))
253elrabsf 3824 . . . . 5 (𝑤 ∈ {𝑦𝐴𝜑} ↔ (𝑤𝐴[𝑤 / 𝑦]𝜑))
2624, 25imbitrrdi 251 . . . 4 (𝑤𝐴 → (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑}))
2726rgen 3061 . . 3 𝑤𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑})
28 wfi 6350 . . 3 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ ({𝑦𝐴𝜑} ⊆ 𝐴 ∧ ∀𝑤𝐴 (Pred(𝑅, 𝐴, 𝑤) ⊆ {𝑦𝐴𝜑} → 𝑤 ∈ {𝑦𝐴𝜑}))) → 𝐴 = {𝑦𝐴𝜑})
291, 27, 28mpanr12 701 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝐴 = {𝑦𝐴𝜑})
30 rabid2 3462 . 2 (𝐴 = {𝑦𝐴𝜑} ↔ ∀𝑦𝐴 𝜑)
3129, 30sylib 217 1 ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  wral 3059  {crab 3430  [wsbc 3776  wss 3947   Se wse 5628   We wwe 5629  Predcpred 6298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299
This theorem is referenced by: (None)
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