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Mirrors > Home > MPE Home > Th. List > wfis2fgOLD | Structured version Visualization version GIF version |
Description: Obsolete version of wfis2fg 6379 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 3795 | . . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑) | |
2 | wfis2fgOLD.1 | . . . . . 6 ⊢ Ⅎ𝑦𝜓 | |
3 | wfis2fgOLD.2 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | sbiev 2313 | . . . . 5 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝜓) |
5 | 1, 4 | bitr3i 277 | . . . 4 ⊢ ([𝑧 / 𝑦]𝜑 ↔ 𝜓) |
6 | 5 | ralbii 3091 | . . 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 6376 | 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 1780 [wsb 2062 ∈ wcel 2106 ∀wral 3059 [wsbc 3791 Se wse 5639 We wwe 5640 Predcpred 6322 |
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-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 |
This theorem is referenced by: (None) |
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