Theorem wl-equsal1t 35700

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 2417, spimt 2386 or sbft 2262, 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 2151	. 2 ⊢ Ⅎ𝑥Ⅎ𝑥𝜑
2		id 22	. 2 ⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥𝜑)
3		biid 260	. . 3 ⊢ (𝜑 ↔ 𝜑)
4	3	2a1i 12	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝑥 = 𝑦 → (𝜑 ↔ 𝜑)))
5	1, 2, 4	wl-equsald 35698	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787

This theorem is referenced by:  wl-equsal1i  35702