Theorem eqsbc3r 3784

Description: eqsbc3 3765 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.)

Assertion

Ref	Expression
eqsbc3r	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴))

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝑉(𝑥)

Proof of Theorem eqsbc3r

Step	Hyp	Ref	Expression
1		eqsbc3 3765	. 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
2		eqcom 2805	. . 3 ⊢ (𝐵 = 𝑥 ↔ 𝑥 = 𝐵)
3	2	sbcbii 3776	. 2 ⊢ ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐵)
4		eqcom 2805	. 2 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
5	1, 3, 4	3bitr4g 317	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  [wsbc 3720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-sbc 3721

This theorem is referenced by:  sbcoreleleq  41241  sbcoreleleqVD  41565