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Mirrors > Home > MPE Home > Th. List > sbralieALT | Structured version Visualization version GIF version |
Description: Alternative shorter proof of sbralie 3348 dependent on ax-ext 2697. (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 3308 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓) | |
2 | 1 | sbbii 2071 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓) |
3 | raleq 3316 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓)) | |
4 | 3 | sbievw 2087 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓) |
5 | cbvralsvw 3308 | . . 3 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓) | |
6 | sbco2vv 2092 | . . . . 5 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ [𝑥 / 𝑦]𝜓) | |
7 | sbralie.1 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
8 | 7 | bicomd 222 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑)) |
9 | 8 | equcoms 2015 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑)) |
10 | 9 | sbievw 2087 | . . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑) |
11 | 6, 10 | bitri 275 | . . . 4 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ 𝜑) |
12 | 11 | ralbii 3087 | . . 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 205 [wsb 2059 ∀wral 3055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1774 df-nf 1778 df-sb 2060 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 |
This theorem is referenced by: (None) |
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