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| Mirrors > Home > MPE Home > Th. List > sbralieALT | Structured version Visualization version GIF version | ||
| Description: Alternative shorter proof of sbralie 3358 dependent on ax-ext 2708, df-cleq 2729, df-clel 2816. (Contributed by NM, 5-Sep-2004.) (New usage is discouraged.) (Proof modification is discouraged.) |
| Ref | Expression |
|---|---|
| sbralie.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| sbralieALT | ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cbvralsvw 3317 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓) | |
| 2 | 1 | sbbii 2076 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓) |
| 3 | raleq 3323 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓)) | |
| 4 | 3 | sbievw 2093 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓) |
| 5 | cbvralsvw 3317 | . . 3 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓) | |
| 6 | sbco2vv 2099 | . . . . 5 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ [𝑥 / 𝑦]𝜓) | |
| 7 | sbralie.1 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 8 | 7 | bicomd 223 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑)) |
| 9 | 8 | equcoms 2019 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑)) |
| 10 | 9 | sbievw 2093 | . . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑) |
| 11 | 6, 10 | bitri 275 | . . . 4 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ 𝜑) |
| 12 | 11 | ralbii 3093 | . . 3 ⊢ (∀𝑥 ∈ 𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 𝜑) |
| 13 | 5, 12 | bitri 275 | . 2 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 𝜑) |
| 14 | 2, 4, 13 | 3bitrri 298 | 1 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 [wsb 2064 ∀wral 3061 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 |
| This theorem is referenced by: (None) |
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