Theorem sbequ2 2291

Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2098. (Revised by BJ, 22-Dec-2020.) (Proof shortened by Wolf Lammen, 3-Feb-2024.)

Assertion

Ref	Expression
sbequ2	⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑))

Proof of Theorem sbequ2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbimp 2099	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		equvinva 2057	. . 3 ⊢ (𝑥 = 𝑡 → ∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦))
3		equcomi 2044	. . . . . 6 ⊢ (𝑡 = 𝑦 → 𝑦 = 𝑡)
4		sp 2225	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
5	3, 4	imim12i 63	. . . . 5 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑡 = 𝑦 → (𝑥 = 𝑦 → 𝜑)))
6	5	impcomd 416	. . . 4 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑))
7	6	aleximi 1859	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → ∃𝑦𝜑))
8	1, 2, 7	syl2im 41	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → (𝑥 = 𝑡 → ∃𝑦𝜑))
9		ax5e 1939	. 2 ⊢ (∃𝑦𝜑 → 𝜑)
10	8, 9	syl6com 38	1 ⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565  ∃wex 1806  [wsb 2097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098

This theorem is referenced by:  stdpc7  2292  sbequ12  2293  sb4a  2518  dfsb1  2519  dfsb2  2531  bj-ssbid2  37169  2pm13.193  45146  2pm13.193VD  45496