Theorem sbrimvlem 2097

Description: Common proof template for sbrimvw 2098 and sbrimv 2306. The hypothesis is an instance of 19.21 2205. (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 2091	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → (𝜑 → 𝜓)))
2		bi2.04 392	. . . 4 ⊢ ((𝜑 → (𝑥 = 𝑦 → 𝜓)) ↔ (𝑥 = 𝑦 → (𝜑 → 𝜓)))
3	2	albii 1827	. . 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 302	. 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
6		sb6 2091	. . 3 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
7	6	imbi2i 339	. 2 ⊢ ((𝜑 → [𝑦 / 𝑥]𝜓) ↔ (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜓)))
8	5, 7	bitr4i 281	1 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071

This theorem is referenced by:  sbrimvw  2098  sbrimv  2306