Theorem 2sbievw 2105

Description: Conversion of double implicit substitution to explicit substitution. Version of 2sbiev 2547 with more disjoint variable conditions, requiring fewer axioms. (Contributed by AV, 29-Jul-2023.) (Revised by Gino Giotto, 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 2104	. 2 ⊢ (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜑 ↔ 𝜓))
3	2	sbievw 2103	1 ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070

This theorem is referenced by:  prtlem5  36011  ichbi12i  43638