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Mirrors > Home > ILE Home > Th. List > tfrlem3-2d | GIF version |
Description: Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.) |
Ref | Expression |
---|---|
tfrlem3-2d.1 | ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V)) |
Ref | Expression |
---|---|
tfrlem3-2d | ⊢ (φ → (Fun 𝐹 ∧ (𝐹‘g) ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem3-2d.1 | . . 3 ⊢ (φ → ∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V)) | |
2 | fveq2 5121 | . . . . . 6 ⊢ (x = g → (𝐹‘x) = (𝐹‘g)) | |
3 | 2 | eleq1d 2103 | . . . . 5 ⊢ (x = g → ((𝐹‘x) ∈ V ↔ (𝐹‘g) ∈ V)) |
4 | 3 | anbi2d 437 | . . . 4 ⊢ (x = g → ((Fun 𝐹 ∧ (𝐹‘x) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹‘g) ∈ V))) |
5 | 4 | cbvalv 1791 | . . 3 ⊢ (∀x(Fun 𝐹 ∧ (𝐹‘x) ∈ V) ↔ ∀g(Fun 𝐹 ∧ (𝐹‘g) ∈ V)) |
6 | 1, 5 | sylib 127 | . 2 ⊢ (φ → ∀g(Fun 𝐹 ∧ (𝐹‘g) ∈ V)) |
7 | 6 | 19.21bi 1447 | 1 ⊢ (φ → (Fun 𝐹 ∧ (𝐹‘g) ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1240 ∈ wcel 1390 Vcvv 2551 Fun wfun 4839 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-iota 4810 df-fv 4853 |
This theorem is referenced by: tfrlemisucfn 5879 tfrlemisucaccv 5880 tfrlemibxssdm 5882 tfrlemibfn 5883 tfrlemi14d 5888 |
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