Theorem fveqeq2d 5584

Description: Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.)

Hypothesis

Ref	Expression
fveqeq2d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
fveqeq2d	⊢ (𝜑 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))

Proof of Theorem fveqeq2d

Step	Hyp	Ref	Expression
1		fveqeq2d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	fveq2d 5580	. 2 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))
3	2	eqeq1d 2214	1 ⊢ (𝜑 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1373  ‘cfv 5271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279

This theorem is referenced by:  fveqeq2  5585  nnnninfeq2  7231  enmkvlem  7263  nninfctlemfo  12361  algcvga  12373  mhmex  13294  resmhm  13319  isghm  13579  lspsneq0  14188  pilem3  15255  2lgslem3c  15572  2lgslem3d  15573  nninfomni  15956