Theorem sbequ2OLD 2251

Description: Obsolete version of sbequ2 2250 as of 3-Feb-2024. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.) Revise df-sb 2070. (Revised by BJ, 22-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbequ2OLD	⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑))

Proof of Theorem sbequ2OLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equvinva 2037	. 2 ⊢ (𝑥 = 𝑡 → ∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦))
2		df-sb 2070	. . . 4 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		equcomi 2024	. . . . . . 7 ⊢ (𝑡 = 𝑦 → 𝑦 = 𝑡)
4		sp 2182	. . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
5	3, 4	imim12i 62	. . . . . 6 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑡 = 𝑦 → (𝑥 = 𝑦 → 𝜑)))
6	5	impcomd 414	. . . . 5 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑))
7	6	alimi 1812	. . . 4 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑))
8	2, 7	sylbi 219	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∀𝑦((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑))
9		19.23v 1943	. . 3 ⊢ (∀𝑦((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑) ↔ (∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑))
10	8, 9	sylib 220	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → (∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑))
11	1, 10	syl5com 31	1 ⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535  ∃wex 1780  [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-12 2177

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

This theorem is referenced by: (None)