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Mirrors > Home > MPE Home > Th. List > sblbisALT | Structured version Visualization version GIF version |
Description: Alternate version of sblbis 2318. (Contributed by NM, 19-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb1.p6 | ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
dfsb1.s4 | ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) |
dfsb1.bi | ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓)))) |
sblbisALT.1 | ⊢ (𝜏 ↔ 𝜒) |
Ref | Expression |
---|---|
sblbisALT | ⊢ (𝜂 ↔ (𝜃 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsb1.p6 | . . 3 ⊢ (𝜃 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) | |
2 | dfsb1.s4 | . . 3 ⊢ (𝜏 ↔ ((𝑥 = 𝑦 → 𝜓) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓))) | |
3 | dfsb1.bi | . . 3 ⊢ (𝜂 ↔ ((𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) ∧ ∃𝑥(𝑥 = 𝑦 ∧ (𝜑 ↔ 𝜓)))) | |
4 | 1, 2, 3 | sbbiALT 2610 | . 2 ⊢ (𝜂 ↔ (𝜃 ↔ 𝜏)) |
5 | sblbisALT.1 | . . 3 ⊢ (𝜏 ↔ 𝜒) | |
6 | 5 | bibi2i 340 | . 2 ⊢ ((𝜃 ↔ 𝜏) ↔ (𝜃 ↔ 𝜒)) |
7 | 4, 6 | bitri 277 | 1 ⊢ (𝜂 ↔ (𝜃 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∃wex 1779 |
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 ax-10 2144 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 |
This theorem is referenced by: sbieALT 2612 |
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