Theorem bj-ssb1 31822

Description: A simplified definition of substitution in case of disjoint variables. See bj-ssb1a 31821 for the backward implication, which does not require ax-11 2021 (note that here, the version of ax-11 2021 with disjoint setvar metavariables would suffice). Compare sb6 2417. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssb1	⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))

Distinct variable group:   𝑥,𝑡

Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ssb1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.21v 1855	. . 3 ⊢ (∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2	1	albii 1737	. 2 ⊢ (∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
3		19.23v 1889	. . . . 5 ⊢ (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
4		equequ2 1940	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
5	4	imbi1d 330	. . . . . . 7 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
6	5	pm5.74i 259	. . . . . 6 ⊢ ((𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
7	6	albii 1737	. . . . 5 ⊢ (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
8		ax6ev 1877	. . . . . 6 ⊢ ∃𝑦 𝑦 = 𝑡
9	8	a1bi 351	. . . . 5 ⊢ ((𝑥 = 𝑡 → 𝜑) ↔ (∃𝑦 𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
10	3, 7, 9	3bitr4ri 292	. . . 4 ⊢ ((𝑥 = 𝑡 → 𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
11	10	albii 1737	. . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
12		alcom 2024	. . 3 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
13	11, 12	bitri 263	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
14		df-ssb 31809	. 2 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
15	2, 13, 14	3bitr4ri 292	1 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  [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:  bj-ax12ssb  31824  bj-ssbssblem  31838  bj-ssbcom3lem  31839