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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ichbidv | Structured version Visualization version GIF version |
Description: Formula building rule for interchangeability (deduction). (Contributed by SN, 4-May-2024.) |
Ref | Expression |
---|---|
ichbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ichbidv | ⊢ (𝜑 → ([𝑥⇄𝑦]𝜓 ↔ [𝑥⇄𝑦]𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ichbidv.1 | . . . . . . . 8 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | sbbidv 2085 | . . . . . . 7 ⊢ (𝜑 → ([𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑦]𝜒)) |
3 | 2 | sbbidv 2085 | . . . . . 6 ⊢ (𝜑 → ([𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑦 / 𝑥][𝑎 / 𝑦]𝜒)) |
4 | 3 | sbbidv 2085 | . . . . 5 ⊢ (𝜑 → ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒)) |
5 | 4, 1 | bibi12d 350 | . . . 4 ⊢ (𝜑 → (([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ 𝜓) ↔ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒 ↔ 𝜒))) |
6 | 5 | albidv 1922 | . . 3 ⊢ (𝜑 → (∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ 𝜓) ↔ ∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒 ↔ 𝜒))) |
7 | 6 | albidv 1922 | . 2 ⊢ (𝜑 → (∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ 𝜓) ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒 ↔ 𝜒))) |
8 | df-ich 44316 | . 2 ⊢ ([𝑥⇄𝑦]𝜓 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ 𝜓)) | |
9 | df-ich 44316 | . 2 ⊢ ([𝑥⇄𝑦]𝜒 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒 ↔ 𝜒)) | |
10 | 7, 8, 9 | 3bitr4g 318 | 1 ⊢ (𝜑 → ([𝑥⇄𝑦]𝜓 ↔ [𝑥⇄𝑦]𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1537 [wsb 2070 [wich 44315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 |
This theorem depends on definitions: df-bi 210 df-sb 2071 df-ich 44316 |
This theorem is referenced by: ichim 44327 |
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