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Mirrors > Home > MPE Home > Th. List > Mathboxes > setinds2f | Structured version Visualization version GIF version |
Description: E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.) |
Ref | Expression |
---|---|
setinds2f.1 | ⊢ Ⅎ𝑥𝜓 |
setinds2f.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
setinds2f.3 | ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑) |
Ref | Expression |
---|---|
setinds2f | ⊢ 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbsbc 3472 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
2 | setinds2f.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
3 | setinds2f.2 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | sbie 2436 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
5 | 1, 4 | bitr3i 266 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 5 | ralbii 3009 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓) |
7 | setinds2f.3 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑) | |
8 | 6, 7 | sylbi 207 | . 2 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) |
9 | 8 | setinds 31807 | 1 ⊢ 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 Ⅎwnf 1748 [wsb 1937 ∀wral 2941 [wsbc 3468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-reg 8538 ax-inf2 8576 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 |
This theorem is referenced by: setinds2 31809 |
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