Theorem sbequivvOLD 2333

Description: Obsolete version of sbequi 2090 as of 7-Jul-2023. Version of sbequi 2090 with disjoint variable conditions, not requiring ax-13 2389. (Contributed by Wolf Lammen, 19-Jan-2023.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbequivvOLD	⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

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

Proof of Theorem sbequivvOLD

Step	Hyp	Ref	Expression
1		equeuclr 2029	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝑦 → 𝑧 = 𝑥))
2	1	imim1d 82	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑧 = 𝑥 → 𝜑) → (𝑧 = 𝑦 → 𝜑)))
3	2	alimdv 1916	. 2 ⊢ (𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑥 → 𝜑) → ∀𝑧(𝑧 = 𝑦 → 𝜑)))
4		sb6 2092	. 2 ⊢ ([𝑥 / 𝑧]𝜑 ↔ ∀𝑧(𝑧 = 𝑥 → 𝜑))
5		sb6 2092	. 2 ⊢ ([𝑦 / 𝑧]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → 𝜑))
6	3, 4, 5	3imtr4g 298	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))

Syntax hints:   → wi 4  ∀wal 1534  [wsb 2068

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 1969  ax-7 2014

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-sb 2069

This theorem is referenced by:  sbequvvOLD  2334