Theorem eqeq1dALT 2741

Description: Alternate proof of eqeq1d 2740, shorter but requiring ax-12 2171. (Contributed by NM, 27-Dec-1993.) (Revised by Wolf Lammen, 19-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
eqeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
eqeq1dALT	⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))

Proof of Theorem eqeq1dALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1d.1	. . . . . 6 ⊢ (𝜑 → 𝐴 = 𝐵)
2		dfcleq 2731	. . . . . 6 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3	1, 2	sylib 217	. . . . 5 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
4	3	19.21bi 2182	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	bibi1d 344	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶)))
6	5	albidv 1923	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶) ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶)))
7		dfcleq 2731	. 2 ⊢ (𝐴 = 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶))
8		dfcleq 2731	. 2 ⊢ (𝐵 = 𝐶 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐶))
9	6, 7, 8	3bitr4g 314	1 ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730

This theorem is referenced by: (None)