Theorem bj-ssb1a 32963

Description: One direction of a simplified definition of substitution in case of disjoint variables. See bj-ssb1 32964 for the biconditional, which requires ax-11 2190. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssb1a	⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → [𝑡/𝑥]b𝜑)

Distinct variable group:   𝑥,𝑡

Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem bj-ssb1a

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . . 6 ⊢ ((𝑥 = 𝑡 → 𝜑) → (∃𝑦 𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
2		19.23v 2023	. . . . . 6 ⊢ (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
3	1, 2	sylibr 224	. . . . 5 ⊢ ((𝑥 = 𝑡 → 𝜑) → ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
4		equequ2 2111	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
5	4	imbi1d 330	. . . . . . 7 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
6	5	pm5.74i 260	. . . . . 6 ⊢ ((𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
7	6	albii 1895	. . . . 5 ⊢ (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
8	3, 7	sylibr 224	. . . 4 ⊢ ((𝑥 = 𝑡 → 𝜑) → ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
9	8	alimi 1887	. . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
10		bj-ssblem2 32962	. . 3 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
11		stdpc5v 2019	. . . 4 ⊢ (∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
12	11	alimi 1887	. . 3 ⊢ (∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
13	9, 10, 12	3syl 18	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
14		df-ssb 32951	. 2 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
15	13, 14	sylibr 224	1 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → [𝑡/𝑥]b𝜑)

Syntax hints:   → wi 4  ∀wal 1629  ∃wex 1852  [wssb 32950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093

This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-ssb 32951

This theorem is referenced by: (None)