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| Mirrors > Home > MPE Home > Th. List > wfis2fgOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of wfis2fg 6377 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 11-Feb-2011.) | 
| Ref | Expression | 
|---|---|
| wfis2fgOLD.1 | ⊢ Ⅎ𝑦𝜓 | 
| wfis2fgOLD.2 | ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | 
| wfis2fgOLD.3 | ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) | 
| Ref | Expression | 
|---|---|
| wfis2fgOLD | ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sbsbc 3792 | . . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑) | |
| 2 | wfis2fgOLD.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜓 | |
| 3 | wfis2fgOLD.2 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | |
| 4 | 2, 3 | sbiev 2314 | . . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝜓) | 
| 5 | 1, 4 | bitr3i 277 | . . . 4 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝜓) | 
| 6 | 5 | ralbii 3093 | . . 3 ⊢ (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 ↔ ∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓) | 
| 7 | wfis2fgOLD.3 | . . 3 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)𝜓 → 𝜑)) | |
| 8 | 6, 7 | biimtrid 242 | . 2 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑)) | 
| 9 | 8 | wfisg 6374 | 1 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 Ⅎwnf 1783 [wsb 2064 ∈ wcel 2108 ∀wral 3061 [wsbc 3788 Se wse 5635 We wwe 5636 Predcpred 6320 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 | 
| This theorem is referenced by: (None) | 
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