Theorem sbi1OLD 2542

Description: Obsolete version of sbi1 2076 as of 24-Jul-2023. Removal of implication from substitution. (Contributed by NM, 14-May-1993.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbi1OLD	⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

Proof of Theorem sbi1OLD

Step	Hyp	Ref	Expression
1		sbequ2 2250	. . . . 5 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → 𝜑))
2		sbequ2 2250	. . . . 5 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑 → 𝜓) → (𝜑 → 𝜓)))
3	1, 2	syl5d 73	. . . 4 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → 𝜓)))
4		sbequ1 2249	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → [𝑦 / 𝑥]𝜓))
5	3, 4	syl6d 75	. . 3 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)))
6	5	sps 2184	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)))
7		sb4OLD 2506	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
8		sb4OLD 2506	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑 → 𝜓) → ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓))))
9		ax-2 7	. . . . . 6 ⊢ ((𝑥 = 𝑦 → (𝜑 → 𝜓)) → ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓)))
10	9	al2imi 1816	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
11		sb2 2504	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) → [𝑦 / 𝑥]𝜓)
12	10, 11	syl6 35	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜓))
13	8, 12	syl6 35	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑 → 𝜓) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜓)))
14	7, 13	syl5d 73	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)))
15	6, 14	pm2.61i 184	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))

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)