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Mirrors > Home > NFE Home > Th. List > sbbi | GIF version |
Description: Equivalence inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
sbbi | ⊢ ([y / x](φ ↔ ψ) ↔ ([y / x]φ ↔ [y / x]ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfbi2 609 | . . 3 ⊢ ((φ ↔ ψ) ↔ ((φ → ψ) ∧ (ψ → φ))) | |
2 | 1 | sbbii 1653 | . 2 ⊢ ([y / x](φ ↔ ψ) ↔ [y / x]((φ → ψ) ∧ (ψ → φ))) |
3 | sbim 2065 | . . . 4 ⊢ ([y / x](φ → ψ) ↔ ([y / x]φ → [y / x]ψ)) | |
4 | sbim 2065 | . . . 4 ⊢ ([y / x](ψ → φ) ↔ ([y / x]ψ → [y / x]φ)) | |
5 | 3, 4 | anbi12i 678 | . . 3 ⊢ (([y / x](φ → ψ) ∧ [y / x](ψ → φ)) ↔ (([y / x]φ → [y / x]ψ) ∧ ([y / x]ψ → [y / x]φ))) |
6 | sban 2069 | . . 3 ⊢ ([y / x]((φ → ψ) ∧ (ψ → φ)) ↔ ([y / x](φ → ψ) ∧ [y / x](ψ → φ))) | |
7 | dfbi2 609 | . . 3 ⊢ (([y / x]φ ↔ [y / x]ψ) ↔ (([y / x]φ → [y / x]ψ) ∧ ([y / x]ψ → [y / x]φ))) | |
8 | 5, 6, 7 | 3bitr4i 268 | . 2 ⊢ ([y / x]((φ → ψ) ∧ (ψ → φ)) ↔ ([y / x]φ ↔ [y / x]ψ)) |
9 | 2, 8 | bitri 240 | 1 ⊢ ([y / x](φ ↔ ψ) ↔ ([y / x]φ ↔ [y / x]ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 [wsb 1648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 |
This theorem is referenced by: sblbis 2072 sbrbis 2073 spsbbi 2077 sbco 2083 sbidm 2085 sbal 2127 sb8eu 2222 pm13.183 2979 sbcbig 3092 sb8iota 4346 |
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