Theorem frege55lem1c 38737

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 3589	. . 3 ⊢ ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵})
2		eqeq1 2775	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
3	2	elabg 3503	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵} → (𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵} ↔ 𝐴 = 𝐵))
4	3	ibi 256	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝑥 = 𝐵} → 𝐴 = 𝐵)
5	1, 4	sylbi 207	. 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 1631   ∈ wcel 2145  {cab 2757  [wsbc 3588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-sbc 3589

This theorem is referenced by:  frege56c  38740