Theorem 2sbievw 2133

Description: Conversion of double implicit substitution to explicit substitution. Version of 2sbiev 2536 with more disjoint variable conditions, requiring fewer axioms. (Contributed by AV, 29-Jul-2023.) Avoid ax-13 2403. (Revised by GG, 10-Jan-2024.)

Hypothesis

Ref	Expression
2sbievw.1	⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
2sbievw	⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ 𝜓)

Distinct variable groups:   𝑥,𝑦,𝜓   𝑥,𝑡,𝑦   𝑦,𝑢

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

Proof of Theorem 2sbievw

Step	Hyp	Ref	Expression
1		2sbievw.1	. . 3 ⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓))
2	1	sbiedvw 2132	. 2 ⊢ (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜑 ↔ 𝜓))
3	2	sbievw 2131	1 ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100

This theorem is used by:  2mos  2676  prtlem5  39718  ichbi12i  48345