Theorem bj-ssbbi 32622

Description: Biconditional property for substitution, closed form. Specialization of biconditional. Uses only ax-1--5. Compare spsbbi 2402. (Contributed by BJ, 22-Dec-2020.)

Assertion

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

Proof of Theorem bj-ssbbi

Step	Hyp	Ref	Expression
1		biimp 205	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
2	1	alimi 1739	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜑 → 𝜓))
3		bj-ssbim 32621	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))
4	2, 3	syl 17	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡/𝑥]b𝜑 → [𝑡/𝑥]b𝜓))
5		biimpr 210	. . . 4 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
6	5	alimi 1739	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ∀𝑥(𝜓 → 𝜑))
7		bj-ssbim 32621	. . 3 ⊢ (∀𝑥(𝜓 → 𝜑) → ([𝑡/𝑥]b𝜓 → [𝑡/𝑥]b𝜑))
8	6, 7	syl 17	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡/𝑥]b𝜓 → [𝑡/𝑥]b𝜑))
9	4, 8	impbid 202	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ([𝑡/𝑥]b𝜑 ↔ [𝑡/𝑥]b𝜓))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  [wssb 32619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839

This theorem depends on definitions:  df-bi 197  df-ssb 32620

This theorem is referenced by:  bj-ssbbii  32624