Theorem sbbiiev 2128

Description: An equivalence of substitutions (as in sbbii 2111) allowing the additional information that 𝑥 = 𝑡. Version of sbiev 2348 and sbievw 2129 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 273	. . 3 ⊢ ((𝑥 = 𝑡 → 𝜑) ↔ (𝑥 = 𝑡 → 𝜓))
3	2	albii 1841	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜓))
4		sb6 2120	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
5		sb6 2120	. 2 ⊢ ([𝑡 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜓))
6	3, 4, 5	3bitr4i 305	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1560  [wsb 2092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-sb 2093

This theorem is referenced by:  sbievw  2129  sbcom3vv  2133  sbcov  2293  sbiev  2348  cbvralsvw  3315