Theorem sbequ2 2246

Description: An equality theorem for substitution. (Contributed by NM, 16-May-1993.) Revise df-sb 2071. (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		df-sb 2071	. . . 4 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2	1	biimpi 219	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		equvinva 2038	. . 3 ⊢ (𝑥 = 𝑡 → ∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦))
4		equcomi 2025	. . . . . 6 ⊢ (𝑡 = 𝑦 → 𝑦 = 𝑡)
5		sp 2180	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
6	4, 5	imim12i 62	. . . . 5 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑡 = 𝑦 → (𝑥 = 𝑦 → 𝜑)))
7	6	impcomd 415	. . . 4 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ((𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → 𝜑))
8	7	aleximi 1839	. . 3 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (∃𝑦(𝑥 = 𝑦 ∧ 𝑡 = 𝑦) → ∃𝑦𝜑))
9	2, 3, 8	syl2im 40	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → (𝑥 = 𝑡 → ∃𝑦𝜑))
10		ax5e 1920	. 2 ⊢ (∃𝑦𝜑 → 𝜑)
11	9, 10	syl6com 37	1 ⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1541  ∃wex 1787  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071

This theorem is referenced by:  stdpc7  2248  sbequ12  2249  sb1OLD  2481  sb4a  2483  dfsb1  2484  dfsb2  2496  bj-ssbid2  34580  2pm13.193  41845  2pm13.193VD  42196