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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-equsal1t | Structured version Visualization version GIF version |
Description: The expression 𝑥 = 𝑦 in antecedent position
plays an important role in
predicate logic, namely in implicit substitution. However, occasionally
it is irrelevant, and can safely be dropped. A sufficient condition for
this is when 𝑥 (or 𝑦 or both) is not free in
𝜑.
This theorem is more fundamental than equsal 2416, spimt 2385 or sbft 2266, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.) |
Ref | Expression |
---|---|
wl-equsal1t | ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfnf1 2155 | . 2 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑 | |
2 | id 22 | . 2 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑) | |
3 | biid 264 | . . 3 ⊢ (𝜑 ↔ 𝜑) | |
4 | 3 | 2a1i 12 | . 2 ⊢ (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))) |
5 | 1, 2, 4 | wl-equsald 35435 | 1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 Ⅎwnf 1791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-12 2175 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 |
This theorem is referenced by: wl-equsal1i 35439 |
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