Theorem 2sbiev 2543

Description: Conversion of double implicit substitution to explicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2410. See 2sbievw 2137 for a version with extra disjoint variables, but based on fewer axioms. (Contributed by AV, 29-Jul-2023.) (New usage is discouraged.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem 2sbiev

Step	Hyp	Ref	Expression
1		nfv 1941	. 2 ⊢ Ⅎ𝑥𝜓
2		2sbiev.1	. . 3 ⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓))
3	2	sbiedv 2542	. 2 ⊢ (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜑 ↔ 𝜓))
4	1, 3	sbie 2540	1 ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  [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  ax-10 2182  ax-12 2219  ax-13 2410

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-sb 2098

This theorem is referenced by: (None)