Theorem bj-ssbcom3lem 31839

Description: Lemma for bj-ssbcom3 when setvar variables are disjoint. Remark: does not seem useful. (Contributed by BJ, 30-Dec-2020.)

Assertion

Ref	Expression
bj-ssbcom3lem	⊢ ([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑦]b[𝑡/𝑥]b𝜑)

Distinct variable group:   𝑥,𝑦,𝑡

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑡)

Proof of Theorem bj-ssbcom3lem

Step	Hyp	Ref	Expression
1		equequ2 1940	. . . . . . 7 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
2	1	imbi1d 330	. . . . . 6 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
3	2	pm5.74i 259	. . . . 5 ⊢ ((𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
4	3	albii 1737	. . . 4 ⊢ (∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
5		19.21v 1855	. . . 4 ⊢ (∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6		19.21v 1855	. . . 4 ⊢ (∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
7	4, 5, 6	3bitr3i 289	. . 3 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
8	7	albii 1737	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
9		bj-ssb1 31822	. . . 4 ⊢ ([𝑦/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
10	9	bj-ssbbii 31813	. . 3 ⊢ ([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑦]b∀𝑥(𝑥 = 𝑦 → 𝜑))
11		bj-ssb1 31822	. . 3 ⊢ ([𝑡/𝑦]b∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
12	10, 11	bitri 263	. 2 ⊢ ([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
13		bj-ssb1 31822	. . . 4 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
14	13	bj-ssbbii 31813	. . 3 ⊢ ([𝑡/𝑦]b[𝑡/𝑥]b𝜑 ↔ [𝑡/𝑦]b∀𝑥(𝑥 = 𝑡 → 𝜑))
15		bj-ssb1 31822	. . 3 ⊢ ([𝑡/𝑦]b∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
16	14, 15	bitri 263	. 2 ⊢ ([𝑡/𝑦]b[𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
17	8, 12, 16	3bitr4i 291	1 ⊢ ([𝑡/𝑦]b[𝑦/𝑥]b𝜑 ↔ [𝑡/𝑦]b[𝑡/𝑥]b𝜑)

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  [wssb 31808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-11 2021

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ssb 31809

This theorem is referenced by: (None)