Theorem 2sbiev 2503

Description: Conversion of double implicit substitution to explicit substitution. (Contributed by AV, 29-Jul-2023.)

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 1896	. 2 ⊢ Ⅎ𝑥𝜓
2		2sbiev.1	. . 3 ⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (𝜑 ↔ 𝜓))
3	2	sbiedv 2502	. 2 ⊢ (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜑 ↔ 𝜓))
4	1, 3	sbie 2500	1 ⊢ ([𝑡 / 𝑥][𝑢 / 𝑦]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  [wsb 2044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-10 2114  ax-12 2143  ax-13 2346

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1766  df-nf 1770  df-sb 2045

This theorem is referenced by:  ichbi12i  43122