Theorem sbequi 2363

Description: An equality theorem for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.)

Assertion

Ref	Expression
sbequi	⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Proof of Theorem sbequi

Step	Hyp	Ref	Expression
1		equtr 1935	. . 3 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
2		sbequ2 1869	. . . 4 ⊢ (𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑 → 𝜑))
3		sbequ1 2096	. . . 4 ⊢ (𝑧 = 𝑦 → (𝜑 → [𝑦 / 𝑧]𝜑))
4	2, 3	syl9 75	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
5	1, 4	syld 46	. 2 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
6		ax13 2237	. . 3 ⊢ (¬ 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
7		sp 2041	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → 𝑧 = 𝑥)
8	7	con3i 149	. . . . 5 ⊢ (¬ 𝑧 = 𝑥 → ¬ ∀𝑧 𝑧 = 𝑥)
9		sb4 2344	. . . . 5 ⊢ (¬ ∀𝑧 𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑 → ∀𝑧(𝑧 = 𝑥 → 𝜑)))
10	8, 9	syl 17	. . . 4 ⊢ (¬ 𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑 → ∀𝑧(𝑧 = 𝑥 → 𝜑)))
11		equeuclr 1937	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
12	11	imim1d 80	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑧 = 𝑥 → 𝜑) → (𝑧 = 𝑦 → 𝜑)))
13	12	al2imi 1733	. . . . 5 ⊢ (∀𝑧 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑥 → 𝜑) → ∀𝑧(𝑧 = 𝑦 → 𝜑)))
14		sb2 2340	. . . . 5 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) → [𝑦 / 𝑧]𝜑)
15	13, 14	syl6 34	. . . 4 ⊢ (∀𝑧 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑥 → 𝜑) → [𝑦 / 𝑧]𝜑))
16	10, 15	syl9 75	. . 3 ⊢ (¬ 𝑧 = 𝑥 → (∀𝑧 𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
17	6, 16	syld 46	. 2 ⊢ (¬ 𝑧 = 𝑥 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
18	5, 17	pm2.61i 175	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  [wsb 1867

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  df-sb 1868

This theorem is referenced by:  sbequ  2364