Theorem frege55lem1c 39768

Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)

Assertion

Ref	Expression
frege55lem1c	⊢ ((𝜑 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑 → 𝐴 = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem frege55lem1c

Step	Hyp	Ref	Expression
1		df-sbc 3712	. . 3 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵})
2		eqeq1 2801	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
3	2	elabg 3607	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵} → (𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵} ↔ 𝐴 = 𝐵))
4	3	ibi 268	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵} → 𝐴 = 𝐵)
5	1, 4	sylbi 218	. 2 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐵 → 𝐴 = 𝐵)
6	5	imim2i 16	1 ⊢ ((𝜑 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑 → 𝐴 = 𝐵))

Syntax hints:   → wi 4   = wceq 1525   ∈ wcel 2083  {cab 2777  [wsbc 3711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-sbc 3712

This theorem is referenced by:  frege56c  39771