Theorem bj-ssbim 31810

Description: Distribute substitution over implication, closed form. Specialization of implication. Uses only ax-1--5. Compare spsbim 2382. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssbim	⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))

Proof of Theorem bj-ssbim

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imim2 56	. . . . 5 ⊢ ((𝜑 → 𝜓) → ((𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜓)))
2	1	al2imi 1733	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
3	2	imim2d 55	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓))))
4	3	alimdv 1832	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) → ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓))))
5		df-ssb 31809	. 2 ⊢ ([𝑡/𝑥]b𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6		df-ssb 31809	. 2 ⊢ ([𝑡/𝑥]b𝜓 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
7	4, 5, 6	3imtr4g 284	1 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))

Syntax hints:   → wi 4  ∀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

This theorem depends on definitions:  df-bi 196  df-ssb 31809

This theorem is referenced by:  bj-ssbbi  31811  bj-ssbimi  31812