Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-findisg | GIF version |
Description: Version of bj-findis 13754 using a class term in the consequent. Constructive proof (from CZF). See the comment of bj-findis 13754 for explanations. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-findis.nf0 | ⊢ Ⅎ𝑥𝜓 |
bj-findis.nf1 | ⊢ Ⅎ𝑥𝜒 |
bj-findis.nfsuc | ⊢ Ⅎ𝑥𝜃 |
bj-findis.0 | ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) |
bj-findis.1 | ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) |
bj-findis.suc | ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) |
bj-findisg.nfa | ⊢ Ⅎ𝑥𝐴 |
bj-findisg.nfterm | ⊢ Ⅎ𝑥𝜏 |
bj-findisg.term | ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏)) |
Ref | Expression |
---|---|
bj-findisg | ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-findis.nf0 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | bj-findis.nf1 | . . 3 ⊢ Ⅎ𝑥𝜒 | |
3 | bj-findis.nfsuc | . . 3 ⊢ Ⅎ𝑥𝜃 | |
4 | bj-findis.0 | . . 3 ⊢ (𝑥 = ∅ → (𝜓 → 𝜑)) | |
5 | bj-findis.1 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜒)) | |
6 | bj-findis.suc | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝜃 → 𝜑)) | |
7 | 1, 2, 3, 4, 5, 6 | bj-findis 13754 | . 2 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → ∀𝑥 ∈ ω 𝜑) |
8 | bj-findisg.nfa | . . 3 ⊢ Ⅎ𝑥𝐴 | |
9 | nfcv 2306 | . . 3 ⊢ Ⅎ𝑥ω | |
10 | bj-findisg.nfterm | . . 3 ⊢ Ⅎ𝑥𝜏 | |
11 | bj-findisg.term | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜏)) | |
12 | 8, 9, 10, 11 | bj-rspg 13561 | . 2 ⊢ (∀𝑥 ∈ ω 𝜑 → (𝐴 ∈ ω → 𝜏)) |
13 | 7, 12 | syl 14 | 1 ⊢ ((𝜓 ∧ ∀𝑦 ∈ ω (𝜒 → 𝜃)) → (𝐴 ∈ ω → 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 Ⅎwnf 1447 ∈ wcel 2135 Ⅎwnfc 2293 ∀wral 2442 ∅c0 3407 suc csuc 4340 ωcom 4564 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-nul 4105 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-bd0 13588 ax-bdim 13589 ax-bdan 13590 ax-bdor 13591 ax-bdn 13592 ax-bdal 13593 ax-bdex 13594 ax-bdeq 13595 ax-bdel 13596 ax-bdsb 13597 ax-bdsep 13659 ax-infvn 13716 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2726 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-nul 3408 df-sn 3579 df-pr 3580 df-uni 3787 df-int 3822 df-suc 4346 df-iom 4565 df-bdc 13616 df-bj-ind 13702 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |