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Mirrors > Home > MPE Home > Th. List > sb3b | Structured version Visualization version GIF version |
Description: Simplified definition of substitution when variables are distinct. This is the biconditional strengthening of sb3 2468. Usage of this theorem is discouraged because it depends on ax-13 2363. (Contributed by BJ, 6-Oct-2018.) Shorten sb3 2468. (Revised by Wolf Lammen, 21-Feb-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb3b | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb4b 2466 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
2 | equs5 2451 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
3 | 1, 2 | bitr4d 282 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 ∃wex 1773 [wsb 2059 |
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-10 2129 ax-12 2163 ax-13 2363 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-ex 1774 df-nf 1778 df-sb 2060 |
This theorem is referenced by: sb3 2468 sb1 2469 |
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