Theorem wl-equsal1i 38232

Description: The antecedent 𝑥 = 𝑦 is irrelevant, if one or both setvar variables are not free in 𝜑. (Contributed by Wolf Lammen, 1-Sep-2018.)

Hypotheses

Ref	Expression
wl-equsal1i.1	⊢ (Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑)
wl-equsal1i.2	⊢ (𝑥 = 𝑦 → 𝜑)

Assertion

Ref	Expression
wl-equsal1i	⊢ 𝜑

Proof of Theorem wl-equsal1i

Step	Hyp	Ref	Expression
1		wl-equsal1i.1	. 2 ⊢ (Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑)
2		wl-equsal1i.2	. . 3 ⊢ (𝑥 = 𝑦 → 𝜑)
3	2	gen2 1829	. 2 ⊢ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑)
4		sp 2222	. . . . 5 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	4	alcoms 2196	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6		wl-equsal1t 38230	. . . 4 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
7	5, 6	imbitrid 247	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑))
8		wl-equsalcom 38231	. . . . 5 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑))
9		wl-equsal1t 38230	. . . . . 6 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ 𝜑))
10	9	biimpd 232	. . . . 5 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → 𝜑))
11	8, 10	biimtrrid 246	. . . 4 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑))
12	11	spsd 2226	. . 3 ⊢ (Ⅎ𝑦𝜑 → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑))
13	7, 12	jaoi 871	. 2 ⊢ ((Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑) → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑))
14	1, 3, 13	mp2 9	1 ⊢ 𝜑

This proof depends on syntax axioms:   → wi 4   ∨ wo 861  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2407

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by: (None)