Theorem sbequiOLD 2534

Description: Obsolete proof of sbequi 2091 as of 7-Jul-2023. An equality theorem for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sbequiOLD

Step	Hyp	Ref	Expression
1		equtr 2028	. . 3 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → 𝑧 = 𝑦))
2		sbequ2 2250	. . . 4 ⊢ (𝑧 = 𝑥 → ([𝑥 / 𝑧]𝜑 → 𝜑))
3		sbequ1 2249	. . . 4 ⊢ (𝑧 = 𝑦 → (𝜑 → [𝑦 / 𝑧]𝜑))
4	2, 3	syl9 77	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
5	1, 4	syld 47	. 2 ⊢ (𝑧 = 𝑥 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
6		ax13 2393	. . 3 ⊢ (¬ 𝑧 = 𝑥 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
7		sp 2182	. . . . . 6 ⊢ (∀𝑧 𝑧 = 𝑥 → 𝑧 = 𝑥)
8	7	con3i 157	. . . . 5 ⊢ (¬ 𝑧 = 𝑥 → ¬ ∀𝑧 𝑧 = 𝑥)
9		sb4OLD 2506	. . . . 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 2030	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
12	11	imim1d 82	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑧 = 𝑥 → 𝜑) → (𝑧 = 𝑦 → 𝜑)))
13	12	al2imi 1816	. . . . 5 ⊢ (∀𝑧 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑥 → 𝜑) → ∀𝑧(𝑧 = 𝑦 → 𝜑)))
14		sb2 2504	. . . . 5 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) → [𝑦 / 𝑧]𝜑)
15	13, 14	syl6 35	. . . 4 ⊢ (∀𝑧 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑥 → 𝜑) → [𝑦 / 𝑧]𝜑))
16	10, 15	syl9 77	. . 3 ⊢ (¬ 𝑧 = 𝑥 → (∀𝑧 𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
17	6, 16	syld 47	. 2 ⊢ (¬ 𝑧 = 𝑥 → (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
18	5, 17	pm2.61i 184	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by: (None)