Theorem wl-equsal1t 32506

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 2279, spimt 2241 or sbft 2367, to which it is related. (Contributed by Wolf Lammen, 19-Aug-2018.)

Assertion

Ref	Expression
wl-equsal1t	⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))

Proof of Theorem wl-equsal1t

Step	Hyp	Ref	Expression
1		nfnf1 2018	. 2 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑
2		id 22	. 2 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3		biid 250	. . 3 ⊢ (𝜑 ↔ 𝜑)
4	3	2a1i 12	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → (𝜑 ↔ 𝜑)))
5	1, 2, 4	wl-equsald 32504	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by:  wl-equsal1i  32508