Theorem sbbiiev 2091

Description: An equivalence of substitutions (as in sbbii 2075) allowing the additional information that 𝑥 = 𝑡. Version of sbiev 2313 and sbievw 2092 without a disjoint variable condition on 𝜓, useful for substituting only part of 𝜑. (Contributed by SN, 24-Aug-2025.)

Hypothesis

Ref	Expression
sbbiiev.1	⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbbiiev	⊢ ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)

Distinct variable group:   𝑥,𝑡

Allowed substitution hints:   𝜑(𝑥,𝑡)   𝜓(𝑥,𝑡)

Proof of Theorem sbbiiev

Step	Hyp	Ref	Expression
1		sbbiiev.1	. . . 4 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))
2	1	pm5.74i 271	. . 3 ⊢ ((𝑥 = 𝑡 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜓))
3	2	albii 1818	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜓))
4		sb6 2084	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
5		sb6 2084	. 2 ⊢ ([𝑡 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜓))
6	3, 4, 5	3bitr4i 303	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1537  [wsb 2063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064

This theorem is referenced by:  sbievw  2092  sbcom3vv  2096  sbcov  2255  sbiev  2313  cbvralsvw  3294