![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sbimi | GIF version |
Description: Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.) |
Ref | Expression |
---|---|
sbimi.1 | ⊢ (φ → ψ) |
Ref | Expression |
---|---|
sbimi | ⊢ ([y / x]φ → [y / x]ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbimi.1 | . . . 4 ⊢ (φ → ψ) | |
2 | 1 | imim2i 12 | . . 3 ⊢ ((x = y → φ) → (x = y → ψ)) |
3 | 1 | anim2i 324 | . . . 4 ⊢ ((x = y ∧ φ) → (x = y ∧ ψ)) |
4 | 3 | eximi 1488 | . . 3 ⊢ (∃x(x = y ∧ φ) → ∃x(x = y ∧ ψ)) |
5 | 2, 4 | anim12i 321 | . 2 ⊢ (((x = y → φ) ∧ ∃x(x = y ∧ φ)) → ((x = y → ψ) ∧ ∃x(x = y ∧ ψ))) |
6 | df-sb 1643 | . 2 ⊢ ([y / x]φ ↔ ((x = y → φ) ∧ ∃x(x = y ∧ φ))) | |
7 | df-sb 1643 | . 2 ⊢ ([y / x]ψ ↔ ((x = y → ψ) ∧ ∃x(x = y ∧ ψ))) | |
8 | 5, 6, 7 | 3imtr4i 190 | 1 ⊢ ([y / x]φ → [y / x]ψ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∃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-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-sb 1643 |
This theorem is referenced by: sbbii 1645 sb6f 1681 hbsb3 1686 sbidm 1728 sbco 1839 sbcocom 1841 elsb3 1849 elsb4 1850 sbalyz 1872 hbsb4t 1886 moimv 1963 oprcl 3564 peano1 4260 peano2 4261 |
Copyright terms: Public domain | W3C validator |