Theorem 2sbiev 2510

Description: Conversion of double implicit substitution to explicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2377. See 2sbievw 2096 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 1914	. 2 ⊢ Ⅎ𝑥𝜓
2		2sbiev.1	. . 3 ⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓))
3	2	sbiedv 2509	. 2 ⊢ (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜑 ↔ 𝜓))
4	1, 3	sbie 2507	1 ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2177  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065

This theorem is referenced by: (None)