Theorem sbrimvlem 2095

Description: Common proof template for sbrimvw 2096 and sbrimv 2305. The hypothesis is an instance of 19.21 2203. (Contributed by Wolf Lammen, 29-Jan-2024.)

Hypothesis

Ref	Expression
sbrimvlem.1	⊢ (∀𝑥(𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))

Assertion

Ref	Expression
sbrimvlem	⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbrimvlem

Step	Hyp	Ref	Expression
1		sb6 2089	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))
2		bi2.04 388	. . . 4 ⊢ ((𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ (𝑥 = 𝑦 → (𝜑 → 𝜓)))
3	2	albii 1823	. . 3 ⊢ (∀𝑥(𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))
4		sbrimvlem.1	. . 3 ⊢ (∀𝑥(𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
5	1, 3, 4	3bitr2i 298	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
6		sb6 2089	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
7	6	imbi2i 335	. 2 ⊢ ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
8	5, 7	bitr4i 277	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  [wsb 2068

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-sb 2069

This theorem is referenced by:  sbrimvw  2096  sbrimv  2305