Theorem eqrdOLD 3623

Description: Obsolete proof of eqrd 3622 as of 1-Dec-2021. (Contributed by Thierry Arnoux, 21-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
eqrd.0	⊢ Ⅎ𝑥𝜑
eqrd.1	⊢ Ⅎ𝑥𝐴
eqrd.2	⊢ Ⅎ𝑥𝐵
eqrd.3	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))

Assertion

Ref	Expression
eqrdOLD	⊢ (𝜑 → 𝐴 = 𝐵)

Proof of Theorem eqrdOLD

Step	Hyp	Ref	Expression
1		eqrd.0	. . 3 ⊢ Ⅎ𝑥𝜑
2		eqrd.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		eqrd.2	. . 3 ⊢ Ⅎ𝑥𝐵
4		eqrd.3	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	biimpd 219	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
6	1, 2, 3, 5	ssrd 3608	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
7	4	biimprd 238	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))
8	1, 3, 2, 7	ssrd 3608	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
9	6, 8	eqssd 3620	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588

This theorem is referenced by: (None)