Theorem sbbiiev 2133

Description: An equivalence of substitutions (as in sbbii 2116) allowing the additional information that 𝑥 = 𝑡. Version of sbiev 2353 and sbievw 2134 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 274	. . 3 ⊢ ((𝑥 = 𝑡 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜓))
3	2	albii 1846	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜓))
4		sb6 2125	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
5		sb6 2125	. 2 ⊢ ([𝑡 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜓))
6	3, 4, 5	3bitr4i 306	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565  [wsb 2097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098

This theorem is referenced by:  sbievw  2134  sbcom3vv  2138  sbcov  2298  sbiev  2353  cbvralsvw  3322