Theorem spsbbi 2073

Description: Biconditional property for substitution. Closed form of sbbii 2076. Specialization of biconditional. (Contributed by NM, 2-Jun-1993.) Revise df-sb 2065. (Revised by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
spsbbi	⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓))

Proof of Theorem spsbbi

Step	Hyp	Ref	Expression
1		biimp 215	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	alimi 1811	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜑 → 𝜓))
3		spsbim 2072	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
4	2, 3	syl 17	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
5		biimpr 220	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
6	5	alimi 1811	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜓 → 𝜑))
7		spsbim 2072	. . 3 ⊢ (∀𝑥(𝜓 → 𝜑) → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜑))
8	6, 7	syl 17	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡 / 𝑥]𝜓 → [𝑡 / 𝑥]𝜑))
9	4, 8	impbid 212	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910

This theorem depends on definitions:  df-bi 207  df-sb 2065

This theorem is referenced by:  sbbidv  2079  sbbid  2246  abbi  2807  sbeqi  38166