Theorem wl-equsal1i 37915

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 1803	. 2 ⊢ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑)
4		sp 2195	. . . . 5 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	4	alcoms 2169	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6		wl-equsal1t 37913	. . . 4 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
7	5, 6	imbitrid 245	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑))
8		wl-equsalcom 37914	. . . . 5 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑))
9		wl-equsal1t 37913	. . . . . 6 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ 𝜑))
10	9	biimpd 230	. . . . 5 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → 𝜑))
11	8, 10	biimtrrid 244	. . . 4 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑))
12	11	spsd 2199	. . 3 ⊢ (Ⅎ𝑦𝜑 → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑))
13	7, 12	jaoi 863	. 2 ⊢ ((Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑) → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑))
14	1, 3, 13	mp2 9	1 ⊢ 𝜑

Syntax hints:   → wi 4   ∨ wo 853  ∀wal 1545  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by: (None)