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Mirrors > Home > ILE Home > Th. List > sborv | GIF version |
Description: Version of sbor 1825 where x and y are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.) |
Ref | Expression |
---|---|
sborv | ⊢ ([y / x](φ ∨ ψ) ↔ ([y / x]φ ∨ [y / x]ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb5 1764 | . . 3 ⊢ ([y / x](φ ∨ ψ) ↔ ∃x(x = y ∧ (φ ∨ ψ))) | |
2 | andi 730 | . . . 4 ⊢ ((x = y ∧ (φ ∨ ψ)) ↔ ((x = y ∧ φ) ∨ (x = y ∧ ψ))) | |
3 | 2 | exbii 1493 | . . 3 ⊢ (∃x(x = y ∧ (φ ∨ ψ)) ↔ ∃x((x = y ∧ φ) ∨ (x = y ∧ ψ))) |
4 | 19.43 1516 | . . 3 ⊢ (∃x((x = y ∧ φ) ∨ (x = y ∧ ψ)) ↔ (∃x(x = y ∧ φ) ∨ ∃x(x = y ∧ ψ))) | |
5 | 1, 3, 4 | 3bitri 195 | . 2 ⊢ ([y / x](φ ∨ ψ) ↔ (∃x(x = y ∧ φ) ∨ ∃x(x = y ∧ ψ))) |
6 | sb5 1764 | . . 3 ⊢ ([y / x]φ ↔ ∃x(x = y ∧ φ)) | |
7 | sb5 1764 | . . 3 ⊢ ([y / x]ψ ↔ ∃x(x = y ∧ ψ)) | |
8 | 6, 7 | orbi12i 680 | . 2 ⊢ (([y / x]φ ∨ [y / x]ψ) ↔ (∃x(x = y ∧ φ) ∨ ∃x(x = y ∧ ψ))) |
9 | 5, 8 | bitr4i 176 | 1 ⊢ ([y / x](φ ∨ ψ) ↔ ([y / x]φ ∨ [y / x]ψ)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∨ wo 628 ∃wex 1378 [wsb 1642 |
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-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-sb 1643 |
This theorem is referenced by: sbor 1825 |
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