Theorem sbrim 2342

Description: Substitution in an implication with a variable not free in the antecedent affects only the consequent. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) Avoid ax-10 2179. (Revised by GG, 20-Nov-2024.)

Hypothesis

Ref	Expression
sbrim.1	⊢ Ⅎ𝑥𝜑

Assertion

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

Proof of Theorem sbrim

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bi2.04 392	. . . . . . 7 ⊢ ((𝑥 = 𝑡 → (𝜑 → 𝜓)) ↔ (𝜑 → (𝑥 = 𝑡 → 𝜓)))
2	1	albii 1852	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑡 → (𝜑 → 𝜓)) ↔ ∀𝑥(𝜑 → (𝑥 = 𝑡 → 𝜓)))
3		sbrim.1	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
4	3	19.21 2246	. . . . . 6 ⊢ (∀𝑥(𝜑 → (𝑥 = 𝑡 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜓)))
5	2, 4	bitri 278	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑡 → (𝜑 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜓)))
6	5	imbi2i 339	. . . 4 ⊢ ((𝑡 = 𝑦 → ∀𝑥(𝑥 = 𝑡 → (𝜑 → 𝜓))) ↔ (𝑡 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜓))))
7		bi2.04 392	. . . 4 ⊢ ((𝑡 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜓))) ↔ (𝜑 → (𝑡 = 𝑦 → ∀𝑥(𝑥 = 𝑡 → 𝜓))))
8	6, 7	bitri 278	. . 3 ⊢ ((𝑡 = 𝑦 → ∀𝑥(𝑥 = 𝑡 → (𝜑 → 𝜓))) ↔ (𝜑 → (𝑡 = 𝑦 → ∀𝑥(𝑥 = 𝑡 → 𝜓))))
9	8	albii 1852	. 2 ⊢ (∀𝑡(𝑡 = 𝑦 → ∀𝑥(𝑥 = 𝑡 → (𝜑 → 𝜓))) ↔ ∀𝑡(𝜑 → (𝑡 = 𝑦 → ∀𝑥(𝑥 = 𝑡 → 𝜓))))
10		dfsb 2101	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ∀𝑡(𝑡 = 𝑦 → ∀𝑥(𝑥 = 𝑡 → (𝜑 → 𝜓))))
11		dfsb 2101	. . . 4 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ∀𝑡(𝑡 = 𝑦 → ∀𝑥(𝑥 = 𝑡 → 𝜓)))
12	11	imbi2i 339	. . 3 ⊢ ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → ∀𝑡(𝑡 = 𝑦 → ∀𝑥(𝑥 = 𝑡 → 𝜓))))
13		19.21v 1972	. . 3 ⊢ (∀𝑡(𝜑 → (𝑡 = 𝑦 → ∀𝑥(𝑥 = 𝑡 → 𝜓))) ↔ (𝜑 → ∀𝑡(𝑡 = 𝑦 → ∀𝑥(𝑥 = 𝑡 → 𝜓))))
14	12, 13	bitr4i 281	. 2 ⊢ ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ ∀𝑡(𝜑 → (𝑡 = 𝑦 → ∀𝑥(𝑥 = 𝑡 → 𝜓))))
15	9, 10, 14	3bitr4i 306	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1816  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-sb 2100

This theorem is used by:  sbiedw  2352  sbied  2538  sbco2d  2547