Theorem bj-ssb1a 33069

Description: One direction of a simplified definition of substitution in case of disjoint variables. See bj-ssb1 33070 for the biconditional, which requires ax-11 2198. (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 2037	. . . . . 6 ⊢ (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
3	1, 2	sylibr 225	. . . . 5 ⊢ ((𝑥 = 𝑡 → 𝜑) → ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
4		equequ2 2123	. . . . . . . 8 ⊢ (𝑦 = 𝑡 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑡))
5	4	imbi1d 332	. . . . . . 7 ⊢ (𝑦 = 𝑡 → ((𝑥 = 𝑦 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜑)))
6	5	pm5.74i 262	. . . . . 6 ⊢ ((𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
7	6	albii 1914	. . . . 5 ⊢ (∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑡 → 𝜑)))
8	3, 7	sylibr 225	. . . 4 ⊢ ((𝑥 = 𝑡 → 𝜑) → ∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
9	8	alimi 1906	. . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
10		bj-ssblem2 33068	. . 3 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → ∀𝑦∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)))
11		stdpc5v 2033	. . . 4 ⊢ (∀𝑥(𝑦 = 𝑡 → (𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
12	11	alimi 1906	. . 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 33057	. 2 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
15	13, 14	sylibr 225	1 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → [𝑡/𝑥]b𝜑)

Syntax hints:   → wi 4  ∀wal 1650  ∃wex 1874  [wssb 33056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105

This theorem depends on definitions:  df-bi 198  df-an 385  df-ex 1875  df-ssb 33057

This theorem is referenced by: (None)