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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-sbievw1 | Structured version Visualization version GIF version |
Description: Lemma for substitution. (Contributed by BJ, 23-Jul-2023.) |
Ref | Expression |
---|---|
bj-sbievw1 | ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 2092 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))) | |
2 | bj-sblem1 34185 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → (∃𝑥 𝑥 = 𝑦 → 𝜓))) | |
3 | sb6 2092 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
4 | ax6ev 1971 | . . . 4 ⊢ ∃𝑥 𝑥 = 𝑦 | |
5 | 4 | a1bi 365 | . . 3 ⊢ (𝜓 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜓)) |
6 | 2, 3, 5 | 3imtr4g 298 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → ([𝑦 / 𝑥]𝜑 → 𝜓)) |
7 | 1, 6 | sylbi 219 | 1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1534 ∃wex 1779 [wsb 2068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-sb 2069 |
This theorem is referenced by: (None) |
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