Theorem sbequOLD 2092

Description: Obsolete proof of sbequ 2090 as of 7-Jul-2023. An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 14-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem sbequOLD

Step	Hyp	Ref	Expression
1		sbequi 2091	. 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
2		sbequi 2091	. . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
3	2	equcoms 2027	. 2 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
4	1, 3	impbid 214	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑))

Syntax hints:   → wi 4   ↔ wb 208  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070

This theorem is referenced by: (None)