Theorem wl-equsal1i 32508

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 1714	. 2 ⊢ ∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑)
4		sp 2041	. . . . 5 ⊢ (∀𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
5	4	alcoms 2022	. . . 4 ⊢ (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6		wl-equsal1t 32506	. . . 4 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))
7	5, 6	syl5ib 233	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑))
8		wl-equsalcom 32507	. . . . 5 ⊢ (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ ∀𝑦(𝑥 = 𝑦 → 𝜑))
9		wl-equsal1t 32506	. . . . . 6 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥 → 𝜑) ↔ 𝜑))
10	9	biimpd 218	. . . . 5 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → 𝜑))
11	8, 10	syl5bir 232	. . . 4 ⊢ (Ⅎ𝑦𝜑 → (∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑))
12	11	spsd 2045	. . 3 ⊢ (Ⅎ𝑦𝜑 → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑))
13	7, 12	jaoi 393	. 2 ⊢ ((Ⅎ𝑥𝜑 ∨ Ⅎ𝑦𝜑) → (∀𝑥∀𝑦(𝑥 = 𝑦 → 𝜑) → 𝜑))
14	1, 3, 13	mp2 9	1 ⊢ 𝜑

Syntax hints:   → wi 4   ∨ wo 382  ∀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-11 2021  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: (None)